Buying Netflix through Apple, but Netflix says I don't have streaming service.

Do you have the Apple TV box? If so, which model? Identify your Apple TV model - Apple Support

You can’t subscribe to Netflix through Apple (anymore). You’ll have to subscribe to Netflix with Netflix themselves directly (i.e. on the web). Once you have the subscription, then you can start the Netflix app on your Apple device, sign in, and link the device to your Netflix account.

Apple TV 2016

That is ambiguous, as two very different models were sold as new in 2016.

1. Apple TV (3rd generation) (model number A1469), 0.9 inch height (2.3 cm), with all pre-configured apps.
2. Apple TV (4th generation) (model number A1625), 1.4 inch height (3.6 cm), with an on-device App Store allowing for custom app installs.

Either way, you will have to sign up with Netflix directly, to get an active Netflix account.

• Some users had reported issues with re-activating expired accounts, especially in combination with Apple TV (2nd or 3rd generation). The solution is then to go to the Netflix site and “update” the email address of the account. The new email can be the same as the old.
• To install the Netflix app on the 4th generation model, you should get it from the on-device App Store. Find apps in the Apple TV App Store - Apple Support

Apple has been charging me a monthly fee for Netflix.

I thought that possibility had ended several years ago.

Is Netflix listed in your Apple subscriptions overview?

If it is, then I would suggest to contact an expert from the Apple Support staff for further assistance. Contact Apple for support and service (phone numbers) or start your support request online (chat/​email/​callback).