Did the Roman Republic really have 60 legions?

TL;DR Possibly, but we cannot be sure.

H. M. D. Parker, "The Roman Legions", Oxford: Clarendon Press 1928, pp. 70-71 goes into some detail, but also points out that an exact accounting is made difficult by the fact that one has to rely on secondary sources.

The final period of the Republic, from 44 B.C. till the battle of Actium, illustrates the same principles of legionary recruiting. Each side raised the troops necessary for its own purposes, and probably had its own system of notation. The secondary sources from which this period is known do not permit of such a detailed investigation into the numbers of the legions employed as the 'Commentarii' of Caesar. To illustrate the extensive conscription it will, however, be sufficient to cite the figures given for the opposing armies at Philippi and Actium. Brutus and Cassius had nineteen, ¹ while their opponents possessed more than forty legions.² At Actium Antony had thirty-one legions, nineteen forming the land army, ³ eight serving on the fleet, and four stationed in Egypt; ⁴ Octavian had between forty and forty-five.⁵ This represents the highest pitch of a system of conscription dictated by the necessities of a civil war. After the battle Octavian was left with an army numbering upwards of sixty legions.
¹ Vell. ii. 65. ² Livy, Ep. cxxii. ³ Plut. Ant. 68. ⁴ Orosius, vi. 19. ⁵ App. B. C. v. 127.

Checking into the cited sources, I am only able to partially confirm the number of legions given by Parker. In the English translation of Velleius reproduced on LacusCurtius, I find:

65 [...] Antony threatened to join forces with Brutus and Cassius, who had now control of seventeen legions, if Caesar rejected this friendly gesture, and said that Caesar was under greater obligations to avenge a father than he to avenge a friend.

This states seventeen legions, not the nineteen referred to by Parker. According to Wikipedia, book 122 of Livius' Epitome (History of Rome), while covering the relevant time period, 43 BC, is no longer extant, so possibly cited indirectly via some other source.

The English translation of Plutarch's Life of Antony found online at LacusCurtius does mention the nineteen legions of the land army:

68 [...] Only a few were aware that Antony had fled, and to those who heard of it the story was at first an incredible one, that he had gone off and left nineteen legions of undefeated men-at‑arms and twelve thousand hors*m*n, as if he had not many times experienced both kinds of fortune and were not exercised by the reverses of countless wars and fightings.

From the English translation of Orosius found online at Attalus, it appears that there were eight legions serving on board Caesar's fleet, not Antony's:

19 [...] On the third day after the battle, Antony transferred his camp to Actium and prepared to decide the issue by a naval engagement. There were two hundred and thirty beaked ships in Caesar's fleet and thirty without beaks, triremes equal in swiftness to Liburnian vessels; eight legions, not counting five praetorian cohorts, were stationed on board the fleet. Antony's fleet had one hundred and seventy ships, but this smaller number was offset by their exceptional size, for in height they stood ten feet above the level of the sea.

The English translation of the Civil Wars section of Appian's Roman History at LacusCurtius does mention forty-five legions at the cited location in book five:

127 Octavian neither pursued Pompeius nor allowed others to do so; either because he refrained from encroaching on Antony's dominions, or because he preferred to wait and see what Antony would do to Pompeius and make that a pretext for a quarrel if he should do wrong (for they had long entertained the suspicion that ambition would bring them into mutual conflict when other rivals were out of the way), or, as Octavian said later, because Pompeius was not one of his father's murderers. He now brought his forces together, and they amounted to forty-five legions of infantry, 25,000 horse and some 40,000 light-armed troops, with 600 war-ships; he had also an immense number of merchant-vessels, which nevertheless he sent back to their owners.

In any event, while the exact number of legions at the end of the Roman Republic may not be known exactly, and may in fact be unknowable unless additional sources from antiquity are found, Octavian's own accounting does indicate that after emerging victorious from the Battle of Actium, he was in command of a massive army that may well have comprised the sixty legions mentioned in various history books. Res Gestae Divi Augusti 3.3 states:

Millia civium Romanorum adacta sacramento meo fuerunt circiter quingenta. Ex quibus deduxi in colonias aut remisi in municipia sua stipendis emeritis millia aliquanto plura quam trecenta et iis omnibus agros adsignavi​ aut pecuniam pro praemis militiae​ dedi.

Octavian claims to have had about 500,000 Roman citizens under his command, of which he dismissed and settled or repatriated 300,000.