If I already have subscriptions for one of the channels (like Acorn) on my iPad, can I use that to watch Acorn on Apple TV?

That depends on the method you want to watch them. If you have an Apple TV HD or 4K then look on the app store and if there is an app for Acorn download it and sign in using the same credentials as you do when you use Acorn on your iPad and it should work just fine.

Apple TV app Channels are accessible on any of your devices.

Streaming services in their own app are accessible on any of your (Apple) devices.

But generally, you can’t exchange subscriptions started within the Apple TV app (Channels) with subscriptions started outside the Apple TV app, or vise versa.

Interesting. I was wondering the same thing. I initiated a subscription to Paramount+ from inside the Apple TV app. I don't know how to login from another device if that device does not support Apple TV. Shouldn't the subscription "stand on its own" and give me the ability to login from wherever i'd like?

I initiated a subscription to Paramount+ from inside the Apple TV app.

Then you will need to watch it in the Apple TV app, on any of your devices that has the Apple TV app.

I don't know how to login from another device if that device does not support Apple TV.

You can’t — unless the originating content provider has made provisions to extend the existing subscription to their administration (and streaming bandwidth). CBS had done so the past, so maybe Paramount+ can do that too (same parent company)?

Shouldn't the subscription "stand on its own" and give me the ability to login from wherever i'd like?

Consider switching by unsubscribing and resubscribing with Paramount+ directly and using the Paramount+ app. In-app Channels are “locked in” to an ecosystem (Prime Video Channels are similar).

Thank you for your answer. This model makes it very easy to subscribe to more channels but it definitely locks you in the AppleTV application. This is probably not a huge deal in general but there are scenarios where you would like to have independence.