In other words, include my title and the name of my employer in my personal email signature in my personal email.
You can, but I wouldn't advise it, for the following reason.
Every last email you send with that information reflects on the company
In other words, if you compose an email that offends someone, and it gets back to your employer, you are out of luck.
It is far too much of a risk to use in anything other than professional correspondence.
It's a bad idea.
It would cause confusion. There are two identities: your personal one, and your work one. These should be kept separate.
If you start sending emails like this and somebody notices that the signature doesn't match the email domain, it will damage both your reputation and your company.
I would myself take this as impersonating somebody else and potential scam attempt.