i wouldn't say nothing because there are a couple of things wrong with using a bitcoin address/key instead of a PGP key.
for starters a bitcoin address is a single-use token, in other words if reusing addresses is already wrong. secondly it requires a centralized database that links identities to addresses which goes against decentralized nature of bitcoin.
and finally this type of signatures make sense to prove ownership of funds not identity and with more complex scripts that we can have (multi-sig, locktime, conditionals, ...) it makes even less sense for identity while it could still be used as "proof of funds".