If we compare this stance to what it meant to be "Roman," it seems that Romans had to be under the rule of the Roman Republic/Emperor, inside Roman territory, and willing to at least give a nominal nod to the divine emperor by making sacrifices occasionally. Many of those under Roman rule weren't really allowed to be Roman citizens, but merely provincial subjects.
"Barbarian" foreigners did adopt aspects of Roman culture, but I don't think the Romans would recognize them as Romans no matter how closely they aped them.
The Greeks didn't have to be part of a particular state or government. They didn't need to be in a particular territory. Any ethnicity was admissible. They just had to adopt the in-group buy ins and be part of the club.
Is this right?
Were the Greeks unique in this open-mindedness regarding their culture?
Great question. I'm sure there will be better answers available but what I would quickly say is that Greek and Roman are slightly different categories.
In the 5th Century BC being Roman was a specific type of citizenship like being Athenian or Spartan. If you were Roman you would probably define your 'ethnicity' as Latin or Italian.
Obviously this idea of 'Romanitas' changes a lot during the Roman empire. Even at this point though, Rome was far more willing to admit others to its citizenship. One great example is the city of Tusculum which fought against Rome, was conquered and made a municipium (self governing city but with a vote in roman elections. In one generation one of its elites was elected consul! This is very very different to Greek city states which were very reluctant to grant citizenship. This liberalness with citizenship was one of the reasons for Rome's rapid expansion through the Italian peninsula and beyond. Granting citizenship was still a roman action though: you couldn't just reclaim yourself roman.
As for 'Greekness' there was no international body to determine Greek ethnicity: there was never one 'Greek state'. Various things contributed to your Greekness: clothes, mythical lineage, political structure but most importantly, language. Another important factor was membership of one of the panhellenic games. During the Pyrrhic wars the Samnites (an italian people) played up their 'Greekness' by talking about a myth where they were descended from spartan settlers. Basically, if it was politically expedient and plausible, it was easy to claim you were Greek. Whether anyone believed you was a different matter...
I would really recommend Nicola Terrenato's 'Early Roman expansion into Italy' for a very important and very readable overview of how Rome was able to co-opt Italian elites. Such a good book and quite cheap as academic books go.
Being 'Hellene' was more closed politically than being 'Roman', as Rome's culture was structured around accepting foreigners and assimilating them politically (for instance, slave gained citizenship on being freed). Roman Republican elites competed to win military victories , which went with bigger armies - hence the ladder from 'Friend of the Romans' to "Ally of the Romans' to 'Latin privileges' to Roman citizen. The assimilation is best seen by the fact that all Italy, Gaul and Spain adopted Latin over the centuries.
Following Alexander, Greek communities sprang up through the east, but membership in the cities was divided between Greek and native. Local elites adopted Greek culture, but how far they were accepted as Greek by Greeks is debatable (eg native Egyptians were not eligible for army service until quite late in Ptolemaic rule). Under Rome, Greek culture very gradually supplanted local cultures in Anatolia and Thrace (Celtic was still spoken by rural people in Galatia in the 5th century, and Isaurian even later). Greek never pushed out local languages in Syria, Egypt or the east.