Did germophobia exist in classical antiquity?

What is germaphobia? It's an obsession, it

"is a pathological fear of contamination and germs. "

If we look for something similar in antiquity we just need to turn that onto its feet: this is about purity or impurity.
Purity rules!
As well for Greeks as for Romans also.

The classical contribution to concepts of contagion and infection thus related less to the individual than to the environment. The term ‘infection’ has a root meaning ‘to put or dip into something’, leading to inficere and infectio, staining or dyeing. This is a further reminder that ‘an infection is basically a pollution’. The same is true not only of ‘contagion’, but also of the noun ‘miasma’, which derives from the Greek verb miaino, a counterpart to the Latin inficere. Impurity is therefore a basic element in all three concepts. These derivations hark back to empirical observation but also evoke the broad spectrum of religious and moral ideas clustering around notions of pollution and taboo. Pollution is concerned not only with time and place, propriety and order, the material and the immaterial, but with the individual’s sense of separateness from his or her environment and how this separateness is to be maintained or regulated.
_{— Alison Bashford & Claire Hooker: "Contagion. Historical and cultural studies", Routledge: London, New York, 2001, p 20.]}

Certain people over-doing it are not that sparse in the literature:

The Superstitious Man
There is a famous character of Greek literature, satirized by the poet Theophrastus, called ‘the superstitious man’:

The danger of pollution is never far from his thoughts. First thing in the morning he washes his hands (perhaps from three springs), and sprinkles his body with lustral water; for the rest of the day he protects himself by chewing laurel. He constantly has his home purified… he declines all contact with birth, death or tombs. He seeks out the Orphotelestai every month, and repeatedly undergoes ablutions in the sea. The mere sight of some poor wretch eating the meals of Hecate [suffering death, disease, destruction] requires an elaborate ritual washing; nor is this enough, but a priestess must be summoned to perform a blood purification.

And all this from a man, Theophrastus, who was himself a Pythagorean vegetarian who must have abhorred meat-eating (and animal clothing), at the very least.
Greek literature is soaked in purity rules and purifications. Such intensity of information certainly makes it look very much as though a ‘cloud of purity rules’ descended on Greece in the fourth and fifth centuries, and subsequent investigations have suggested that new words, and new temple equipment, were indeed imported into Greek culture just prior to this time; but we know that the ancient cosmology of purification was already well established throughout Eurasia, and it is perhaps better to see not an intensification but a fragmentation of this tradition in Greece.
This hypothetical superstitious man was certainly caught up in Orphism, a fifth-century Greek sect known for its onerous ascetic requirements. The followers of Orpheus formed what is known as a ‘mantic’ cult, deriving from the prophetic traditions of seers and shamans, and their wandering seers or healing priests (telestai) would sing beautiful hymns and incantations over the sufferer, prescribing herbs, charms, and a pure new way of life through chastity, vegetarianism, white garments, and the ecstatic worship of Dionysus–Bacchus. Theophrastus meant to imply that the purifications of the superstitious man were excessive, or at least extremely scrupulous by average standards—sufficient even for a sanctified priest.
_{— Virginia Smith: "Clean. A History of Personal Hygiene and Purity", Oxford University Press: Oxford, New York, 2007, p85–86.}

It might be argued that Greek scientific medicine was in itself a form of this phobia:

The word for the dirt that caused disease was miasma (from miaino, to pollute, via the root mia- meaning defilement or destruction); and miasma could be generated in any place at any time, for whatever divine reason. When it reached the earthly world, however, it was specifically associated with foul airs, waters, and places. Greek scientific disease theory suggested that macrocosmic disease pollution came via certain airborne miasmin—germlike ‘seeds of disease’ wafting down from the outer universe in billowing clouds of polluted air that were immanently poisonous and contagious. Whatever the miasma touched on contact with the microcosm it tainted, and then spread itself steadily through the healthy living material ‘like the dyeing or staining of a cloth’. There was no obvious distinction made between macrocosmic miasma and microcosmic contagion, although this was a distinction that greatly concerned physical scientists from the seventeenth to the early twentieth centuries.

The Greeks were obsessed with this kind of purity. But they also noted that they were not unique in this respect and not even the first among peers of the time:

When the well-travelled Herodotus assumes the role of an anthropologist of religion and draws parallels between Greeks and Egyptians, he observes that while the Greeks are generally very concerned with ritual purity, they are far surpassed in this by the Egyptians. For Herodotus, the Egyptians are the most god-fearing of all people, and the nation most obsessed with purity: the Egyptians are, in his words, ‘religious beyond measure’; he catalogues their purity practices, and remarks in admiration that ‘their religious observances (threskeiai) are innumerable’.
_{— Andrej Petrovic & Ivana Petrovic: "Inner Purity and Pollution in Greek Religion. Volume I: Early Greek Religion", Oxford University Press: Oxford, New York, 2016, p26.}

Herodotus On The Egyptians In The Fifth Century B.C.: “They always wear freshly washed linen clothes. They make a special point of this. They have themselves circumcised for reasons of cleanliness, preferring cleanliness to a more attractive appearance. Priests shave their bodies all over every day to keep off lice or anything else dirty.” [Ashenburg]

The Greek term for religious pollution is miasma, commonly translated as ‘stain’ or ‘defilement’; a person affected by such pollution is labelled with a cognate adjective miaros (‘stained’, ‘defiled’, ‘polluted’). The noun miasma is never used to denote physical dirt, but rather signifies ritual impurity that can be dangerous and contaminating. Miasma is understood as dangerous because it compromises human communication with the divine and renders rituals ineffective or, in the worst case, downright sacrilegious. Some types of miasma contaminate through contact. Death, for instance, pollutes a whole house and its inhabitants, and purification and exclusion from the shrine for a fixed period of days is necessary for everyone affected. Even a visitor to the house who is not a member of the household may be rendered polluted for a certain number of days. [Petrovic, p 36.]

Physical pollution through bodily fluids, sex or corpses was one part of the miasma that spread through contagion, but could be overcome by (symbolic) removal of pollutants/temporary abstinence.

But also compared to actual Greek habits, all later Roman bath habits must have looked like acute over-purification (and frivolity). While Gibbon links this Roman preference for warm and hot baths to the decline of Rome's Empire, they certainly did not think it a problem:

“Baths, wine, and sex ruin our bodies, but they are the essence of life—baths, wine, and sex.”
_{— Epitaph on the tomb of Titus Claudius Secundus, first century}

While the contemporary Greeks were a bit more 'spartan':

“Swiftly, safely, sweetly” was the motto of Asclepiades of Bithynia, who popularized Greek medicine in Rome in the first century B.C. and who preferred bathing his patients to bleeding them — hence his motto. He was a great advocate of cold baths in particular and was known as “the Cold Bather.”
_{both quotes — Katherine Ashenburg: "The Dirt on Clean: An Unsanitized History", Vintage Canada: Toronto, 2008 (e).}

This means it is always very dependent on your point of view to describe a certain practice or 'level of purity' as either sub-standard or excessive. Gibbon's 19th century verdicts about the virtues of cold bathing or swift showers when he would come to judge current American habits with long, hot showers are not difficult to imagine.

One example to drive home the importance of context to contemporary observers is again found in a well known drama:

The audience knows that Hippolytus’ purity is under threat, and, based on this knowledge, his wish for ritual purification cannot be interpreted as obsessively and excessively puritan, but as a justified self-defence mechanism.
_{— Hippolytus’ Purity under Triple Threat: Phaedra, the Nurse, and Theseus [Petrovic, p 200–, here 222.]}

If you look at the Bible and do not skip the first part there are many, many rules pertaining to purity, ritual purity and corporeal purity as well. And, as usual, these beliefs were quite wide spread in the ancient orient:

These Elchasites were termed ‘baptisers’ by the collator of the Greek Cologne Mani Codex, and may also be identified with the group known to later Arab observers as al-Mughtasila (The Cleansers). These designations point to the sect’s most defining practices, their constant ritual ablutions, which ranged from personal bathing up through baptisms for the vegetables they ate. It was these relentless baptisers who served as the central formative influence for Mani, who stayed among them for the next twenty years of his life.
According to the testimony of Mani’s companions recorded in the Greek Cologne Mani Codex, Mani ultimately broke with the Elchasites over ritual practice, especially the constant purification that defined them to outsiders. Responding to Elchasite critics after his split from the group, Mani told the story that the waters had themselves rebuked the founder of their sect for his ritual bathing. In Mani’s story, Elchasai’s bathing pool took the form of a man and said, “Is it not enough that your beasts abuse me? Yet you [yourself ] maltreat [my home] and commit sacrilege [against my waters].” Elchasai’s efforts to find a more genial place to bathe were answered with further criticism: “We and those waters of the sea are one. Therefore you have come to sin and abuse us.” Like Elchasai, Mani claimed to have had his own visitations from the waters, among other spiritual visitors who taught him the basic precepts of his new faith, Manichaeism.
_{— [Cynthia Kosso and Anne Scott: "The Nature and Function of Water, Baths, Bathing, and Hygiene from Antiquity through the Renaissance", Brill: Leiden, Boston, 2009, Ch.: Scott John McDonough: “We and Those Waters of the Sea are One”: Baptism, Bathing, And The Construction Of Identity In Late Ancient Babylonia" p 264.]}

The modern definition of germaphobia (mysophobia) is too dependent on the knowledge of the existence of germs. If we look at the essence of its meaning, the fear, sometimes obsessive, of contamination, defilement and impurity, then this concept might be seen as very widespread in antiquity, with the caveat that ancient and modern observers might have quite different ideas about the concrete meanings applied to those who were or are observed.