The boundary of our self is constantly spreading outwards onto the things around us. Seems to be part of human nature.
So no, it's not bad for new activities to become part of your identity. It can be bad for them to become your whole identity, because it leaves you really vulnerable to outside attacks on your self-conception.
(On a side note, it seems to me that a lot of the Silicon Valley startup mythology is focused on encouraging low self-complexity and an obsessive focus on external success. Now that I re-read some of PG's early essays, several of them seem actively harmful to one's mental health. The YC application used to ask you "How are you an 'animal'?", in reference to an early essay where he suggested that successful startup founders often act like caged animals - as if denying your humanity is "success".)
It's people who work at normal jobs that he calls caged animals, and founders are the wild animals.
"In fact, getting a normal job may actually make you less able to start a startup, by turning you into a tame animal who thinks he needs an office to work in and a product manager to tell him what software to write."
"One of the best tricks I learned during our startup was a rule for deciding who to hire. Could you describe the person as an animal? It might be hard to translate that into another language, but I think everyone in the US knows what it means. It means someone who takes their work a little too seriously; someone who does what they do so well that they pass right through professional and cross over into obsessive."
It used to be of course that you were a Ford man or a Chrysler man. But nowadays it's become much rarer.
I agree that it's natural, but I think that the targets for identification shift over time and I don't think it's random.