Possibly what you’re looking for is ‘single sign on’ or SSO.
There’s a plugin called WP Auth Server which provides SSO between WordPress websites without the need for any extra sign-on service. Maybe it’ll do what you want. I don’t have first hand experience with this or SSO in WordPress but have set it up in other contexts.
Connect your app to WordPress or use SSO to connect multiple websites
with the same username and passwords. No 3rd party servers are needed
with WP OAuth Server. Everything you need is in this plugin.