What are the implications of Lean not having the type `Set`?
In Coq Set behaves basically like the smallest possible Type. Unlike Coq however, Lean lets you select the "level" of the type manually, such as Type 9 (or Sort 10), something which is not allowed in Coq - meaning that you do not need Set in Lean, you can just use Type 0 (or just Type, or Sort 1) instead as it is basically the same thing.
I should also note that in Lean Prop is the same as Sort 0 (or Sort) (which does not have large elimination) and Type n (for n : nat) is the same as Sort (n + 1) (which has large elimination).

TL;DR: Type (with no "arguments") is the Set of Lean