iTunes not recognizing authorized computer

I'm trying to back up an older iPhone to my new laptop. It backs up but then wants to authorize the computer to transfer purchases. I agree and it then says I can't authorize more than 5 computers. I went into my apple iCloud account and removed all pcs. There are 3 MacBooks and multiple iPhones, iPads etc. also authorized. When I signed in on the new pc, it re-added the new pc and I can see it in the account. I can also see the same laptop referred to as an iPhone (presumably because the computer synced to iCloud). But iTunes still insists I need to add this computer and that I already have five computers authorized.

How do I get iTunes to recognize that it's the same device.