In other words if Alice has a wallet that contains the address xyz and Bob has a wallet that contains the same address xyz
both have the same private key?
(And thus all individual addresses of Alice and Bob can be signed with their own or the other one's private key?)
Well it is slightly more complex. It is possible that Alice and Bob have different private keys which produce the same address because addresses are a hash of the public key. If that happens (essentially 0% chance) either private key can sign a transaction to spend coins sent to the address in question.