Yes and no.
1. Since studying is never free, you're still technically paying for studying abroad, just not anything extra.
2. You obviously still have to arrange everything there like housing, travel, etc.
Also
3. Not all degrees/university allow it
4. Even if they do, not all degree consider that semester/year as an integral part of your cursus (as in, they allow you to spend a semester/year in another european country but you don't get to have that semester/year counted towards the validation of your degree)
It's a super nice thing that keeps getting bigger and definitely helps to forge a "european identity" in some ways, but across a generation it's still a minority that gets to be able to experiment it, to give some perspective for a single year you have 2.6M students in France, in 30y around 610k were able to experiment Erasmus.
As for the original question, identity is always multi faceted, you start from yourself and gradually go up from there, I'm myself, I identify as a member of my family (thus my name), I then have roots which are defined by a town/region (eg. extended family past and present, for me Bretagne) above that is the state that defines my nationality (I'm french) and above there's this shaky notion of European that we're trying to build up on.
It's a long process and there's a lot of things to work on but I can definitely see a foreseeable future in a few decades where "European" will be an even stronger sentiment shared accross countries there.
And of course we're all earthlings, that does not bear much meaning now but centuries in the future who knows (though maybe by that time we will have transcended our physical bodies and that would once again change everything)