It must be doable. Though so far I believe only the opposite has been done - using an existing website login system with Flarum. See the Wordpress SSO and Laravel Passport login extensions.
How did you expect to sync the login ?
Flarum isn't an oauth server out of the box but I think it shouldn't be too complicated to make it work as one.
Or if both apps have access to the same database / session storage you could just store the cookies in a way both apps can access and check Flarum sessions validity from you ASP.NET app.
EDIT: as it's not a part of core you will probably need to create a Flarum extension to do something like my oauth suggestion. You can check the developer resources under the dev tag and let us know if you have any issue !
Oh, and almost forgot, welcome to the community ?