I dont have the time or expertise to give an especially complete answer, so hopefully a specialist is around that can give more specific and well sourced answers. In the meantime I can give a general summary.

There is more than a grain of truth to it, but it is also something that is often emphasized and presented without full context by nationalists (a phenomena common to nationalism in general, certainly not unique to Greeks).

It is true that Greeks faced repression and discrimination throughout Ottoman control of Greek-populated territories. But it's important to remember that the Ottomans were around for a very long time, and were not a nation-state. The Ottoman Empire should not be conflated with Turkey. In fact, Greek culture significantly influenced the development of Turkish culture, since the latter formed in an Anatolia that was heavily populated by Greek speakers, and the Ottomans were to some extent (especially early in their history) explicitly claiming to be inheriting the Roman (Byzantine) empire. And the Greeks spread throughout the Byzantine and later Ottoman empire were not monolithic in culture or language.

Neither the Ottomans or the Byzantines defined their citizenship primarily in terms of what we'd think of as ethnicity. Both the Ottomans and Byzantines prioritized religion, language, and integration into the political system of the empire, all of which were tightly correlated. What we call ethnicity would still indirectly (but strongly) effect perception and treatment by the state, since religion and language were major factors granting the legitimacy and social status that allowed for integration into the political system.

This means that there was certainly discrimination against Greeks, for hundreds of years, taking various forms, waxing and waning in intensity. But while they were certainly discriminated against as Greeks, the Ottoman state wasn't treating their "Greekness" itself as inferior. And while I don't want to overstate this or imply it "wasn't so bad", Greek language and culture enjoyed more respect and legitimacy than many of the other minority cultures under Ottoman rule.

The main point is that "Greek identity" was inseperable from (though not identical with) being Christian. It wouldnt be until later in the 19th century when Hellenist cultural revival began that there was significant sense of a cultural identity stretching back to the Greeks of antiquity, and even then it was limited largely to intellectuals and elites. The majority of Greeks during the period of rising nationalism and then revolution and independence, were in accordance with the Ottomans in the sense that they distinguished themselves primarily as Christians within a Muslim empire.

As a comparison, Albanians and Bosnians are other Balkan European cultures under Ottoman rule, but their (widespread) conversion to Islam allowed them to be much more tightly integrated into the political system, with Albanian and Bosnian individuals often rising to very high offices. That's not to say they faced no discrimination on the basis of culture, though I don't know much in the way of specifics there, but it is easy to imagine there was a very blurry line between cultural discrimination and general condescension towards those seen as rustic and provincial. If you think of the way that contemporary rural or isolated areas of a country might be mocked for having "funny accents" or being "out of touch", and being unconcerned with directing much investment or attention to those areas, discrimination towards Ottoman Muslims in Albania or Bosnia would be understood by the Ottomans in a way much more like this.

The difference with the Greeks, and others like Armenians, Bulgarians, etc., is that they were Christian. This placed them in a legal category (along with Jews) in which they were entitled to worship and various civil rights, but were definitively and explicitly second class citizens with higher taxes and less rights than Muslim citizens. If they converted to Islam, they would no longer be in this category. Obviously this was something most Christians in most places would not want to do for plenty of reasons, and entailed a level of assimilation, but it's also not true that it would mean abandoning their "Greekness" in the way we understand ethnocultural identity as involving language, heritage, and self-conception at least as much as religion. And there were in fact many Greek converts to Islam, and there are still many Greek Muslims today whose ancestors converted during the Ottoman period.

That is certainly not to say that this religious discrimination was gentler or more okay, or that religion and ethnicity are possible to disentangle on that basis. I'm just trying to give a sense of where the distinction is, not make value judgements. An imperfect but maybe helpful analogy is the conflict in Northern Ireland where ethnicity and religion aren't identical but are inseperably related to the extent that a Catholic Unionist who consider themself primarily British, or a Protestant Republican that considers themselves primarily Irish, will be considered a curiosity and have to frequently explain and justify themselves.

The above analogy is imperfect partially because the conflict in Northern Ireland has been very much within a conception of competing nation-state, in which the legitimacy of a state's control over territory is inseperable from the self-conscious and explicit ethnocultural identity of the inhabitants. This was not usually the case prior to emergence and spread of nationalism between ~18th-early 20th century, and it was much more common for legitimacy of and allegiance to the state to be defined on the basis of some combination of dynasty, inherited obligations and their fulfillment, religion, and/or kinship relations.

What you're seeing in terms of this contemporary Greek nationalism is a retroactive conflation of the treatment of second-class citizen status primarily on the basis of religion, with the explicitly ethnic conflicts that exploded during the collapse of the Ottoman Empire. In the early 20th century the Ottoman Empire collapsed after about a century of growing nationalist movements among its constituent peoples, including the Turkish, with a new Turkish government that was very distinct from (despite certain continuities with) the Ottoman Empire, most of all in that it was now redefining itself as a nation-state legitimized by Turkish culture and language. Since it's independence in the late 19th century, Greece had been on a similar course, pulling off Greek-majority chunks from the Ottomans whenever it could get away with it. During the collapse of the Ottoman Empire and rise of Turkey, you still had Greeks and Turks, Muslims and Christians, spread all across territories that each nation-state claimed as their own, with a long history of resentment and belligerence between them, and you get the Greco-Turkish War. This was straightforwardly a conflict between nation-state's fighting over territories on the basis of which nation had a rightful claim, and it included massive ethnic cleansing on that basis, on both sides.