Although I don’t know it myself, others often suggest Miami of Ohio as a school with a similar vibe, though of course, not Catholic. University of Dayton (Catholic) also gets good reviews as a school with a strong community, similar to ND.
Oops, I just saw the interest in engineering. Don’t know anything about Miami of Ohio or Dayton in that regard. Lehigh and Bucknell both strong schools, and again, although I haven’t visited them personally, I hear that they are nice campuses, with strong communities. Both are smaller than ND and have with greek life – and while ND does not have greek life, its residential halls are often described as fraternity-sorority like, with philanthropy, parties, rivalries, intense identification with your own hall etc.