If it were MIT or Stanford then prestige might matter a bit, but both are ABET accredited for mechanical engineering. Both are also “fit” schools that are only appropriate for strong students who want to work very hard for a full 4 years without let up in a tough environment where they suddenly became average the day they showed up on campus.
Otherwise, ABET accreditation is going to matter more. Engineers really do not care about “prestige”. They care about whether whatever you designed actually works, or if it falls down.
One rare exception might be if you wanted to get a bachelor’s degree in engineering, then go to law school, and then work in investment banking or management consulting (possibly on legal issues somehow related to engineering). This seems like such an unlikely niche thing that I would just forget about it.
And yes, an engineering graduate is going to find themselves working alongside other engineers who graduated from a huge range of other universities, and as long as you all graduated from an ABET accredited school no one is going to care where you went to university.