Can I recover an old Xbox 360 account without an email or the console?

Since it's an online account and not offline, you could recover it without a 360 console, but you would need the email.

So you can have multiple accounts on an xbox 360 or xbox one and they can be on both at the same time but you can only log into one of them at the same time. Now to recover both onto a new console you will need to seperately recover each account. Each one
will have a different email and password, and if a gamer tag shows up that is different to the one on the xbox 360 that means that you may have entered the wrong email and you'll need to just double check that on the old console as the emails will be stored
there.

There is no way you would have been able to remove the email from the account without adding a new one to the account.

If your not sure what email is on the account you can get ahold of xbox support by going to support.xbox.com and if they can verify ownership of the account then they can tell you what email is associated with the gamertag.

Or you can try the old email and go to account.live.com/acsr and try and recover the account.

If you dont remember the email that was used for the account then there will be no option to get the account back. Try this soloution: https://support.xbox.com/en-US/my-account/microsoft-account/forgot-microsoft-account-solution or you could try to contact
the support and get an call back from them. Hope that helps.

I doubt they can with serial numbers. They would usually ask for your answers to the verification questions you added when you created the account.