How to open A Link in a Browser and automatically login the user into the opened Client?

I have an UI (written in Vue) and I want to open a link to another Client under a different domain but same realm, and automatically Login the User there.
For that I want to use the SSO feature.

Now in the context of on Client, everything is fine. Once I logged in, I can open multiple Browser tabs with that client.

If I want to open another client in another Browser tab, I still get a login page.
If I click login with keycloak it also does not ask me for username or password, since I am logged in.
But I don’t want I button to show, I want to have SSO for all Clients, so I only expect one Login page for all these Clients.

As far as I know I am using openID Connect, and have at least 2 Keycloak Clients running.
One is the UI I am writing and the other is the target application.

So assuming all these clients support either oidc or saml: Is this behavior “day-dreaming”?
If not how can I achieve that?

Note that you cannot set the Bearer Token as a authorization header, when opening a link in another tab.

Sorry I am a bit new to configuring Keycloak, so I am a bit new to those protocols as well.
For me I only came across Keycloak when dealing with its tokens and using its Libraries for UIs and Node.js Servers.