My e-mail is forwarded by a private ISP to my Thunderbird account.

There is no such thing as a 'Thunderbird account'. Thunderbird is an email client you use to access the email account you set up with an email provider, e.g. Gmail.

Can I "un-configure" the Gmail account for my Thunderbird identity within Thunderbird and, if so, how?

I have no idea what you're talking about.

Okay. So, Thunderbird, my "email client" receives all my mail a private provider. (Is "provider" the correct term?). And while I do have and have had for some time, two Gmail accounts, neither of them are for the identity I use in Thunderbird.

I am not an IT person so be patient if I have difficulty asking the right questions. I do appreciate your help.

Can you describe what the actual problem is? What are you doing? What happens? And what do you expect to happen?

Sure. I wanted to sign in to one of my Gmail accounts yesterday and used the "sign in with a different account" link - that's when I noticed that my Tbird identity was suddenly one of my options for signing in to Gmail. I have noticed as well that I see duplicate (synchronized I guess) e-mails in one of my Gmail accounts that match yet another identity in Tbird.

Now, I may very well have brought this on myself (gigo is always true) but I have tried and have no idea how to undo the synchronization between Google and Tbird.

Does that help? Basically what I want to do is keep Google out of my business within Mozilla -

... used the "sign in with a different account" link - that's when I noticed that my Tbird identity was suddenly one of my options for signing in to Gmail.

This has got nothing to do with Thunderbird. You may have set up another identity within Gmail, which happens to be your non-Gmail account. I'd start looking in your Gmail account settings (on the Google web site, not Thunderbird account settings).

It doesn't explain the duplicate e-mails though. It's like the two accounts are synchronized. Also, I keep an address book so when I set up a new account I can enter the user name, e-mail associated with it and my password and I only have the two original Gmail accounts entered here. If I had deliberately set up a separate account I would have entered it in my password keeper.

I'll just keep at it - thanks for your help.

Deleted all my Gmail accounts. Still don't understand what happened but no big - thanks again for your help and patience.

To keep different email identities completely separate requires a little extra effort in setting up accounts, since there is a default to set up one SMTP output address and send everything out of that, no matter what mailbox received messages. You have to set a separate outgoing server and a separate reply-to address.