Email signatures are essentially end to end. The company would have to keep a file of all the identities of all the people that everyone in the company knew. To make this useful, the company would also have to record information about who knew who.
As already mentioned, the email might also be encrypted (the most common case). Then the signature would not be available as it would be hidden under the encryption. These sorts of filtering issues are why corporate cybersecurity people, somewhat ironically, tend to dislike encrypted email. There is an opportunity here to make better email clients that treat encrypted, but unsigned, emails with suspicion.
As already mentioned, the email might also be encrypted (the most common case). Then the signature would not be available as it would be hidden under the encryption. These sorts of filtering issues are why corporate cybersecurity people, somewhat ironically, tend to dislike encrypted email. There is an opportunity here to make better email clients that treat encrypted, but unsigned, emails with suspicion.