It does, but it's not crazy to say that personal identity barely exists outside of group identity. Sure, some groups are less defining than others, but from what I gather the general consensus is that there is no such thing as 'personal identity' that doesn't also overlap with group identity. We're social creatures.

Of course, some people might have a more defined personal identity than others, but those are outliers. The vast majority of humans define their self based on groups that are important to them (with parents/family usually being dominant, followed by friends and culture).

Basically, I'd argue there is not much self left if you take away the different group identities.