Currently, when Unread message icon is enabled, gmail will display the total number of unread emails in your inbox.
I personally cannot be bothered to clear out my inbox, so this number becomes completely useless to me.
Is there a way change this badge to display the number of new emails since the tab was last opened, or maybe even just unread emails from Today?
Nope, it only checks your inbox folder for all unread messages. Your only options to use this specific icon feature involve managing your inbox
You could
Brute force it: Select > All > Select all conversations in primary > Mark as read
Set up a rule/script to archive old items or move them to another folder
Or install an extension that supports the kind of filtering your looking for
