Yes. Being "Dorian" or "Ionian" was very much a matter of ethnicity. Aside from inheritance -- whether one or the other was "in your blood", so to speak -- they also involved a variety of customs, fashions, style of dress, religious festivals, myths, architectural styles and other artefacts, and of course political allegiance.

But it certainly wasn't just a matter of which ethnic group you were born into! Cities could swap from one to the other if there was good reason to do so. For example, in Sicily, the city of Gela adopted Dorian customs while Himera didn't. Herodotos 8.73 tells us that the city of Kynoura in Lakonia, which was initially Ionian, became "dorianised" under Argive rule; and Megara became Dorian under Peloponnesian rule.

For further reading you can't do better than Jonathan Hall's Ethnic Identity in Greek Antiquity (Cambridge, 1997), from which the above examples are cribbed.