Google says that it doesnt support my version of Firefox? But when I look at the version that I am using it says this is the up to date version

Hi, you're using Firefox 115 ESR, which is the latest version supported on this device (macOS 10.12).

The newest Firefox 152 needs macOS 10.15 or later.

As I understand, Google doesn't support Firefox 115 anymore. So you could try to install User-Agent Switcher and spoof your browser "user-agent" string to something newer.

Can the version of Firefox that I am using support Google Gmail. which says it should be updated to ESR which I am using I think?

ESR is not tied to one version. The older Firefox 115 ESR is based on the old Firefox 115.0 Release but with security and stability fixes since. It was originally going to end around Fx 115.15.0 esr or so but Mozilla decided to keep it updated so the EOL Windows 7, 8, 8.1 and macOS 10.12, 10.13, 10.4 OS users can have a secure web browser for a while still as Chrome/Chromium dropped support early 2023..

The current ESR is Firefox 140.0 ESR (currently 140.12.0esr and updated till 140.15.0esr) and the next major ESR will be based on the Firefox 153.0 Release on July 21.

You cannot use these ESR versions as Firefox 116.0 and later requires macOS 10.15 (ten.fifteen) or later to run.

This is basically the same as support.mozilla.org/questions/1570596 — the consensus is their nag bars are only cosmetic, and don't seem to really reflect any issue with functionality whatsoever so you should still be able to use the services as before.