I’d just emphasize this really depends on several possible variables: how AP credits work, how the general curriculum works, and how many courses you need for the second major.
What could in theory be a variable, but isn’t usually in practice, is the course requirements for the engineering majors, because as others explained that is not much of a variable for engineering specifically.
And then in terms of favorable curriculums, again others already explained how Rochester’s sort of exploration-in-depth form of distributional requirements is helpful, intentionally in fact.
Another college with a similar curriculum structure is WashU. A bit more complicated in description, but the upshot is the same, between the two majors you will already be satisfying most of your graduation requirements, and it won’t be hard usually to do the rest. They also tend to be on the light side in terms of course requirements for their majors, and again that is intentional, they really like kids to be able to do multiple unrelated majors (FYI, my S24 at WashU declared two majors, Computational Biology and Classics).
OK, so that gives you an idea of what you can look for.
And finally, I agree the degree name you put on this second interest tends not to really matter. Like if you can do a certain number of credits that gets you to a minor but not a second major at your school, this is unlikely to really matter to your future. It is more about just knowing the curriculum won’t be forcing you to take a lot of classes you don’t really want to take.
ETA: By the way, to my knowledge almost all US engineering colleges still have some form of gen ed requirements, you cannot in fact do 100% of your courses in engineering. So that can’t possibly be the expectation for competitive engineering graduates in the US.
On the other hand, I understand doing the bare minimum to graduate sometimes might not be ideal, although I think here you can actually lean on the above point about engineering requirements. Like, ABET or such will not typically let you get a specific type of engineering degree without you taking a robust list of specific courses on top of the general engineering courses they require.
This is why it can be hard to actually do an engineering degree and a second full major and fulfill your gen eds all in four years . . . unless either you are using a lot of APs to cover gen eds, or the gen eds are designed such that a second full major could cover most of them. But I would not automatically assume it is impossible for you to do a competitive amount of engineering courses and do a second major IF at least one of those conditions is met.