Did the Roman Empire have penal colonies?

It seems that the term "penal colony" would be evoking quite modern, if not 'Australian', imagery.

When we look at Roman sources, not that much springs to mind.

True: They frequently sent people into exile, often to islands.

That sounds more like Napoleon on Elba or St. Helena, compared to what a "penal colony" would describe now. And even that is contested by comparing Patmos now with 2000 years ago:

A common misconception in commentaries and popular prophetic writings is that the island of Patmos, where John was exiled, was a sort of Alcatraz (Swindoll 1986:3) or St. Helene where Napoleon was exiled (Saffrey 1975:392). This is partly due to 19th-century travelers who described the island as “a barren, rocky, desolate-looking place” (Newton 1865: 223) or as “a wild and barren island” (Geil 1896: 70). Unfortunately, these 19th-century perceptions are not accurate in describing the island in John’s day.
_{The King and I: Exiled To Patmos, Part 2}

Of course, one option of punishment would have been more like the former in our imagination, being enslaved and send into mines and quarries, if not just killed. Roman prison being quite temporary affair in nature. Some skepticism regarding the veracity of this bit of Catholic tradition of calling Patmos a "penal colony" seems warranted.

But purely about the exiles, Tacitus writes:

[3.68] Tiberius, that his proceedings against Silanus might find some justification in precedent, ordered the Divine Augustus's indictment of Volesus Messala, also a proconsul of Asia, and the Senate's sentence on him to be read. He then asked Lucius Piso his opinion. After a long preliminary eulogy on the prince's clemency, Piso pronounced that Silanus ought to be outlawed and banished to the island of Gyarus. The rest concurred, with the exception of Cneius Lentulus, who, with the assent of Tiberius, proposed that the property of Silanus's mother, as she was very different from him, should be exempted from confiscation, and given to the son.

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30 1 When members then expressed the view that Serenus should be punished according to ancestral custom,18 he sought to mitigate the odium by interposing his veto. A motion of Asinius Gallus, that the prisoner should be confined in Gyarus or Donusa, he also negatived: both islands, he reminded him, were waterless, and, if you granted a man his life, you must also allow him the means of living. Serenus was, therefore, shipped back to Amorgus. And since Cornutus had fallen by his own hand, a proposal was discussed that the accuser's reward should be forfeited whenever the defendant in a charge of treason had resorted to suicide before the completion of the trial. The resolution was on the point of being adopted, when the Caesar, with considerable asperity and unusual frankness, took the side of the accusers, complaining that the laws would be inoperative, the country on the edge of an abyss: they had better demolish the constitution than remove its custodians. Thus the informers, a breed invented for the national ruin and never adequately curbed even by penalties, were now lured into the field with rewards.

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15.71 Rome all this time was thronged with funerals, the Capitol with sacrificial victims. One after another, on the destruction of a brother, a kinsman, or a friend, would return thanks to the gods, deck his house with laurels, prostrate himself at the knees of the emperor, and weary his hand with kisses. He, in the belief that this was rejoicing, rewarded with impunity the prompt informations of Antonius Natalis and Cervarius Proculus. Milichus was enriched with gifts and assumed in its Greek equivalent the name of Saviour. Of the tribunes, Gavius Silvanus, though acquitted, perished by his own hand; Statius Proximus threw away the benefit of the pardon he had accepted from the emperor by the folly of his end. Cornelius Martialis, Flavius Nepos, Statius Domitius were then deprived of the tribuneship, on the ground, not of actually hating the emperor, but of having the credit of it. Novius Priscus, as Seneca's friend, Glitius Gallus, and Annius Pollio, as men disgraced rather than convicted, escaped with sentences of banishment. Priscus and Gallus were accompanied respectively by their wives, Artoria Flaccilla and Egnatia Maximilla. The latter possessed at first a great fortune, still unimpaired, and was subsequently deprived of it, both which circumstances enhanced her fame.

Rufius Crispinus too was banished, on the opportune pretext of the conspiracy, but he was in fact hated by Nero, because he had once been Poppæa's husband. It was the splendour of their name which drove Verginius Flavus and Musonius Rufus into exile. Verginius encouraged the studies of our youth by his eloquence; Rufus by the teachings of philosophy. Cluvidienus Quietus, Julius Agrippa, Blitius Catulinus,

[Note] Petronius Priscus, Julius Altinus, mere rank and file, so to say, had islands in the Ægean Sea assigned to them Cædicia, the wife of Scævinus, and Cæsonius Maximus were forbidden to live in Italy, their penalty being the only proof they had of having been accused. Atilla, the mother of Annæus Lucanus, without either acquittal or punishment, was simply ignored.
_{Tacitus, Ann. 3.68; 4.30; 15.71}

Take note that Tacitus mentions Donusa, Gyarus, and Amorgus, but not Patmos. It seems likely that Patmos was not among the most prominent places – if we still want to compare them to "penal colonies" – of banishment then, known to Tacitus.

To reiterate: How these islands compared to one another as places of exile in the conditions present seems to vary a lot:

The island (Latin: Gyaros or Gyara) also served as a place of exile during the early Roman Empire. Writing in the early 2nd century AD, the Roman historian Tacitus records that, when Silanus, the proconsul of the province of Asia was accused of extortion and treason, and it had been proposed in the Roman Senate that he be exiled to Gyaros, the Roman Emperor Tiberius allowed him to be sent to the nearby island of Kythnos instead, since Gyaros was "harsh and devoid of human culture" (Annales 3.68-69). When confronted with another recommendation to exile a defendant to Gyaros, Tiberius once more declined, noting that the island was deficient in water, and that those granted their lives ought to be granted the means to live (4.30). The defendant was allowed to go into exile on Amorgos instead. The Roman poet Juvenal, a near-contemporary of Tacitus, mentions this island twice in his Satires: first as a place of exile for particularly vile criminals (1.73), and second as a symbol of claustrophobic imprisonment (10.170). In the second reference, Juvenal compares the restlessness of Alexander the Great to that of a man imprisoned:

One globe was not enough for the youth from Pella,
He seethed within the narrow confines of the world,
as if he were hemmed in by the cliffs of Gyara or by tiny Seriphos.

Thus, from early republican times, this form of punishment was quite a peculiar institution, and curiously for the high strata of society, that is Roman citizens:

Therefore, the people often judge crimes punishable by a fine when the defendants have held the highest office, and the people alone judge capital cases. Concerning the latter, they have a practice which is notable and deserves mention. Their custom allows those on trial for capital offenses the freedom to depart openly when found guilty, thus sentencing themselves to voluntary exile, even if only one of the “tribes” has not yet given their verdict. There is safe refuge for these exiles in Neapolis, Praeneste, Tibur, and other states which have treaties with the Romans.
Plb. 6.14.6–8.

The normal order of events in a case involving exile was consistent throughout Roman Republican history. When accused of a crime, a defendant could quit Roman jurisdiction and seek the safety of exile. He could flee before trial commenced or wait until the completion of legal proceedings before departing. Based on Polybius’ statement, the accused could leave a iudicium populi (trial before the comitia centuriata, a citizen-assembly) anytime before the last “tribe” had cast its vote. In other words, he was free to seek exilium before he was formally convicted. If the trial was before a iudicium publicum (jury court), however, the defendant could even wait until after conviction before deciding on flight. The city of Rome was off-limits to all exiles. Italy was added to this restricted territory sometime after he Social War in the first century. Any community holding the Roman citizenship probably could not be entered legally by exiles, although no source specifically states this.8 After he had left proscribed territory, the fugitive could go where he wished. Once a Roman quit his homeland and went into exile, the concilium plebis (plebeian citizen assembly) generally passed a decree of aquae et ignis interdictio. This plebiscite formally prohibited the fugitive from returning to the Roman state. Thus many banished Romans chose to become citizens of a new community. Interdiction from fire and water also imposed some quasi-legal penalties on the fugitive, most notably the forfeiture of property.

As Cicero points out in the Pro Caecina, unlike other states, the Romans had no laws employing banishment as a penalty. For a Roman citizen, exile was a method of avoiding punishment. Due to this practice of allowing the accused to flee Roman jurisdiction, there are very few cases in our extant sources of the death penalty being carried out against a condemned criminal. Thus, as I have previously mentioned, exilium was the practical outcome of nearly all capital trials in the Republican period. (p 17–22)
_{Gordon P. Kelly: "A History of Exile in the Roman Republic", Cambridge University Press: Cambridge, New York, 2006. DOI/gBooks}

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The number of imperial exiles whose names are known is not large. The majority are men and women of wealth and influence. Among them one finds members of the imperial family, senators with Republican leanings, government officials, ladies of rank, imperial freedmen fallen from favor, orators and literary men, philosophers, and teachers. Humble folk are seldom mentioned by name in the sources. About half of the exiles whose names I have found belong to the reigns of Tiberius and Nero, for which we have nearly complete accounts in Tacitus' Annals. Of forty-six persons stated to have been exiled under Nero, at least half may be regarded as political exiles.
The mildest form of banishment involved relegation from Rome, or from Rome and Italy, or from a province, either for a term of years or for life, without a designated place of residence, and (in the case of citizens) without loss of civil rights. For example, Dio Chrysostom, banished under Domitian from Rome and Italy and from Bithynia, the province of his birth, traveled widely in the Roman empire during his fourteen years of exile. Plutarch, writing his consolatory essay on exile to a man who was thus free to travel about, reminded him of the joys of travel, of the possibility of going if he wished to Eleusis for the mysteries, to Delphi for the Pythian games, etc. He also advised choice of the best and most pleasant city as a place of residence. Few cities which were the deliberate choice of imperial exiles are known. Helvidius Priscus, banished from Italy by Nero, spent his exile at Apollonia, a university town in Illyria.
_{Mary V. Braginton: "Exile under the Roman Emperors", The Classical Journal, Vol. 39, No. 7 (Apr., 1944), pp. 391-407. (jstor)}

Regarding the information we might glean relating to specifically John and Patmos:

Patmos is a small island in the Aegean, some 40-50 miles south-west of Ephesus, volcanic in character. It was used, according to Pliny (nat hist. iv, 23) as a place of exile, and so we must suppose that it was as a political exile, or rather as a victim of religious persecution, that John found himself there. With this the ancient tradition agrees—Tertullian (depraem. 36), Clement of Alexandria, Origen, Eusebius {H.E. in, 18), and Jerome. Now there were various grades of punishment. A man could be made 'servus poenae' and condemned, for example, to work in the mines. No doubt the conditions of working as a slave in a Roman mine would be conducive to apocalyptic visions, but they would leave no leisure for writing them down at the time, or much prospect of survival to record them later. And in any case Patmos does not seem to have had any mines. It was, however, used for the less drastic punishments of 'deportatio' and 'relegatio'. The former involved loss of civil rights and forfeiture of property, while the latter involved only compulsory residence in a designated area, to leave which was a capital offence. Tertullian speaks of John as 'in insulam relegatus': he had been a lawyer, and may be assumed to use the term correctly. We have no means of judging the value of the tradition on which he relied, but it seems reasonable to accept its veracity.

'Relegatio' was a punishment reserved for 'honestiores', provincials as well as citizens, except when it was meted out to a whole class of persons, as, for example, in Claudius's expulsion of the Jews from Rome. Thus the poet Ovid was 'relegated' from Rome to Tomi on the Black Sea, and Herod Antipas to Lugdunum in Gaul — cf. Josephus, Ant. XVIII, 252, αυτον δε φυγη αιδιω εζημιωσεν ατοδειξας οικητηριον αυτου Λυγδουνον τολιν της Γαλλιας – Flavia Domitilla to Pontia (Eusebius, H.E. in, 18). The point of 'relegatio' was to remove a person far from his old associations and so keep him out of mischief.

From this, three pieces of evidence may be inferred about John and his exile in Patmos: (1) the scene of the crime for which he was expelled can hardly have been as close to Patmos as Ephesus-Jerusalem, Alexandria or Rome (so Tertullian) are possible, but he probably never set foot in Ephesus until his release from Patmos; (2) John was 'honestior', a member at least of the Jewish aristocracy—which presumably means a Sadducee; (3) if his offence was preaching the Gospel, he must have suffered banishment before there was any precedent clearly established for making the preaching of Christianity a capital offence. As St Paul was presumably executed in the early sixties on the charges brought against him in Acts xxiv. 5 as ανδρα ... λοιμον και κινοθτα στασεις τασιν τοις ᾽Ιοθδαιος τοις κατα την οικουμενην τρωτοστατην τε της των Ναζωραιων αιρεσεως. John may even have been sentenced before Paul's execution—certainly before the Neronian persecution.
He is said by Eusebius to have returned from exile after the death of Domitian (A.D. 96) (H.E. in, 23), and thus may have lived in exile over thirty years. He chose to go to Ephesus upon his release as being the nearest centre of Christian life. Any connexions he may once have had with his home would have been severed in the interval—particularly if it had been from Judaea that he originally came.
_{JN Sanders: "St John on Patmos", New Testament Studies, Volume 9, Issue 02, January 1963, pp 75–85. DOI}

Another note on terminology to consider when reading ancient sources for the period in question:

The major ancient sources […] Needless to say, they do not agree. […] The English term 'exile' may serve to translate a number of Latin terms beyond the Latin exsilium. It can refer to the legal penalties of deportatio (capital exile, in which the exile loses his Roman citizenship) and relegatio (in which the exile retains his Roman citizenship), both of which developed during the Principate; it may also be used to translate the Latin aquae et ignis interdictio (banishment from fire and water), which, as will be described later, is not in origin a legal penalty. Exsilium itself may refer to voluntary exile or to banishment, and is not a word used only in legal contexts; it is, however, part of a technical phrase, solum vertere exilii causa, which refers to a Roman's decision to take up the citizenship of a new city, usually as the result of a criminal trial.
_{Sarah T. Cohen: "Augustus, Julia and the Development of Exile 'Ad Insulam'", The Classical Quarterly, New Series, Vol. 58, No. 1 (May, 2008), pp. 206-217. (jstor)}