From your description it appears you are not signed up for AppleTV+.
On your iPhone go into Settings>Your Name(at the top)>Subscriptions. Do you see AppleTV+ within your active subscription list? Do you see any badging offering AppleTV+ free 3 months?
If you do see the AppleTV+ offer then follow it to sign up for the service.
If you see AppleTV+ under your active descriptions, I suggest checking the AppleID login used on your AppleTV. Be sure it is the same as your iPhone.
If you do not see either the offer or it under the active subscriptions, I suggest reaching out to Apple Support:
Subscriptions and Billing - Official Apple Support
Be aware these offers may not apply if, for example, you've used an offer for AppleTV+ previously.