Apple ID keeps saying my password is incorrect even after changing passwords

Is this with an iPhone or your Apple TV (since I see you use "Apple TV" in your user name)

If it's an iPhone you should also make sure your App Store sign-in, which is separate from your main Apple ID iCloud sign-in, is signed into the Apple ID you think it is. You can check this under Settings > iCloud > Media & Purchases (iOS 14) or Settings > iCloud > iTunes & App Store (iOS 13). And you can sign out and back in there for good measure.

If that doesn't help this makes me think that the apps were installed under a different Apple ID. If these aren't paid apps and you don't care about any data associated with the apps (saved games, etc.) you should try deleting the app entirely then reinstall it from the App Store so the app from that point will be entirely associated with your currently signed-in Apple ID.

If this is indeed your Apple TV you should still do the same thing I recommended, delete your apps (https://support.apple.com/en-us/HT200126) and then reinstall them once you are 100% sure you are signed in properly and that it's just the App being associated with another Apple ID and unable to update because of that.

Gotcha. So it's not something that Apple is going to consider to be a bug. Although some apps will continue to work fine even when downloaded (regardless of if paid, or free) with a previous Apple ID, no app will update through the App store under a different Apple ID. And certainly as you experienced, at least in the past, if you put in the correct Apple ID that the particular app required that would still get you updated.

So if it's a free app you should make sure you're signed in with your main Apple ID for Media & Purchases (iTunes/App Store), delete the app, and reinstall it. If it's a paid App I would delete it, make sure you're signed in with your old Apple ID for Media & Purchases, then reinstall it. Then switch back to your main Apple ID for Media & Purchases.

The only other clever option would be to set up a Family Sharing Group under your old Apple ID (as the head of the "family") and make sure you have App Sharing. Depending on the developer of that particular App they may or may not allow their App to be shared that way. This is the cleanest solution but again, the App developer controls whether they want to allow or disallow that for their app.

If the first method doesn't work, and the Family Sharing option doesn't work, then the only option would be to repurchase the app under your main/current Apple ID.

Family Sharing information.

Family Sharing.         

Family Sharing - Set up.