Adding two public keys secp256k1 (the signature scheme used by Bitcoin) has the interesting property that you can add two public keys to get a combined key that requires both private keys to produce signatures for.

Is there another, more widely used public key cryptography scheme with the same property?
Yes.  Discrete-log based cryptosystems (e.g., El Gamal) have a similar property.  It's "multiply", not "add", but that's just because that's the group operation for discrete-log based cryptosystems.

More generally, I suggest you look at threshold cryptography.  Threshold public-key cryptosystems are a class of systems that can achieve the sort of thing you mention.