of course linux was created to compete with other unixes. it's literally baked into the name. competition transcends 'product' as you seem to be using it, which seems to be something akin to 'formal market offering'. linus meant to create an OS for a segment that he thought was underserved (OS hobbyists). it competed from the beginning with other OSes for mindshare and usage (but not necessarily for end-use). so you're really just distinguishing the segment served, not competition (other than to say it likely won't be competitive outside its niche).

more than that, i don't think this framing does you any good with the segment this browser is targeting, as developers still want others to recognize their (good) work. but that's fine, as there's plenty of time to develop the brand (and yes, brand also exists outside of being a formal product).