Apple TV update requires iPhone

Do you mean the TV app on your iPhone? The Apple TV app version will always match the OS version, for being bundled software. The 2016 iPhone SE (1st generation) can be updated to iOS 15.7.

Or do you mean the Apple TV app on a third party device? iPhone is optional, not mandatory.

Or do you mean updating the system software on an Apple TV box? iPhone is optional, not mandatory.

I see. It seems that two-factor authentication is halting you. Two-factor authentication requires iOS 9 or later (≈iPhone 4S or later), so your phone’s age shouldn’t be the issue by a long stretch. TFA does require the matching Apple ID account on the secondary device. It won’t work for different accounts.

You may request a new fresh verification code at any time from Settings on your iPhone, or any other trusted device for this Apple ID account.

Two-factor authentication for Apple ID - Apple Support

Get a verification code and sign in with two-factor authentication - Apple Support

… wants me to update my Apple ID

Is that the exact message text? I have never heard of something similar mentioned before, except for new accounts that were incomplete or unverified.

If I misinterpreted or if the issue persist, 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).

Apple TV HD (4th generation) should ask when it wants the verification code.

Older models Apple TV (2nd or 3rd generation) may need a special procedure – if the model applies to you.

Identify your Apple TV model - Apple Support

Else 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).

Do contact the Apple Support staff about this anomaly.