Re:" iTunes Student discount and free AppleTV, old AppleTV subscription payment still going on strong"
This is Apple's webpage about the Student offer:
https://offers.applemusic.apple/student-offer
For Apple TV:
webpage says:
"Enjoy 1 month free then a special student rate of $5.99/month1".
Lower down it explains the 1 at the end of above sentence:
- "New subscribers only. Plan automatically renews at $5.99/month until cancelled.
College students only. Verification required. Terms apply:"
2, :Limited-time offer; offer may end at any time. Free Apple TV access for students ends when you no longer qualify or do not renew your Apple Music Student subscription. Offer good for verified college students only and does not extend to a Family Sharing group"
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Linked Wepage above also says:
"Want to learn more?
Click here to learn about Apple Music’s Student Plan eligibility, how to sign up for the Apple Music Student plan, how to access Apple TV, and how to reverify your student status."
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Seems, you are a long term Apple TV subscriber ... and eligibility for the offer is for new subscriber students.
Am no expert, just an Apple user, like yourself. If you want verification:
If you are with Unidays, SheerID, or AliPay you would reach out to them.
Else:
Apple support's Billing and Subscriptions Support Team have a website (info / support / contact options etc)
Subscriptions and Billing - Official Apple Support
Another article has tips for changing an existing plan: I don't know whether you can change your Apple TV plan to student without being a new subscriber: How to switch to a different subscription plan - Apple Support