Apple Family Sharing is not available for many third party streaming services; same for Paramount+ through the Paramount+ app. (For some services, Family Sharing requires changing the subscription plan to a higher tier.)
As Mac doesn’t have an Paramount+ app, you would be using web access instead — but only if you subscribed with Paramount+ directly. Paramount+ app subscriptions can be billed through Apple (through your Apple ID), or can be billed through Paramount+ directly, depending on how you signed up.
Apple Family Sharing should be available for Apple TV app in-app Channels. Such would list as “[service X] tv channel” in your subscriptions overview. Access the in-app Channels through the listed activated/added channels in the sidebar of the TV app on Mac. “Connected apps” do not apply to the Mac edition of the TV app; that only applies to iPhone, iPad, Apple TV box, where those separate apps are available.
Finding TV shows through Search may lead to the Store (buy) version instead.
I can’t check if Paramount+ offers their service as in-app channel in the UK at this time.
How can I avoid this in future?
When the app prompts you to pay for content, and you agree to that, then expect the resulting billing. Not sure why you expected a different outcome. In-app channels that are shared wouldn’t prompt you for a payment.