How can I resolve another Apple ID appearing when updating an app in the App Store?

Apps are permanently tied to the apple account used to purchase them.

That app was apparently purchased using a different account from the one you are using now.

Delete it and re-purchase/download it using the ID you currently use on the device.

Did you change your Apple ID at some point? Or did you set up an iPhone which wasn't prepared correctly for a new user? Apps are forever tied to the Apple ID they were dowloaded under. The only way to update the App would be to sign out of the current Apple ID and sign back in under the Apple ID the app was downloaded under. If you don't know the Apple ID the app was downloaded under this is a concern as it is highly likely you may get locked out of the phone at some, with no way to get back in, which is why I asked if you set up a phone which wasn't properly prepared for you as the new user?

If it is just a case of having forgotten the Apple ID you used previously and you are the original owner of the phone, delete the App from your iPhone, then download it again under your new Apple Account.