Ironically, the Firefox version of this extension requests permissions to inspect all domains, not just Gmail -- indicating that the extension itself may be performing tracking.
I can't tell offhand whether the Chrome version does the same.
I can confirm that in chrome it accesses only gmail.com and in general we can set access permissions to extensions on Chrome[1], for Ugly Email I can see that accessing other domains are disabled by default.
As for Firefox, the reviews seem to tell otherwise, it's not been updated for over 2 years (same for chrome as well).
[1]https://uglyemail.com/