What other webapps do you want to log into?
If your website is custom built, and/or if you were planning to use something like the Laravel framework, consider building the site directly on top of Flarum. Flarum may be known as a community framework, but it's essentially a complete app framework just like many other PHP frameworks.
If the apps are third-party, a good starting point would probably be something based on oauth. Wasn't there an existing Flarum oauth server implementation at some point?
Implementing a global login/logout without the need to click a button on each different app/subdomain might or might not be possible depending on how much control you have to customize your other apps.
