more stringently --> more tightly. (I don't know which one is proper in this case.)
I think we need to use "each other" over there, because it makes a difference in the meaning. I mean, though we use "us", we may mean to direct attention to somebody else. For example, we can say "What made us connect our family/our friends, etc.". But what is especially meant there is "to connect each other".
And about "similarities of ours".. "ours" is used to indicate something that belongs to us. If we say "similarities of ours", here what will "ours" stand for? I really don't get it. :/
As I said before, the line is a bit weird and hard to render.
I think "tightly" would be better.
What about: "What made us connect with one another..."?
And..."these similarities of ours" I suppose that means "the similarities (that exist) between us" Am I right?
If so, perhaps we can say: "these similarities between us" or simply "our similarities".