I just spoke with Apple Support and got quite a bit of a runaround. At first it seemed positive.
I have two devices that both started having this problem at the exact same time. I've tried both devices across 3 different wifi networks, and one of them on a mobile network (one of them is wifi only). My iCloud account is the only one that has ever been logged in to either device.
They had me test downloading new apps. Deleting and then downloading an app that wouldn't update. Both were fine. But updating existing apps still didn't work.
One thing they also suggested was signing out of iTunes, resetting the network, then going in to App Store and only signing back in when prompted after updating the app. All failed.
This person then had me sign in to iTunes on my phone using someone elses account. This worked. To me this confirms the problem is my account, not my devices.
They put me through to the iTunes Account Billing Team, they had me try a few times while they watched my account. They couldn't find any issues. Put me back through to Technical Support.
This person really struggled to understand the problem. Kept insisting I was trying to use two accounts despite the fact the ONLY time a second account has ever touched my phone was earlier today while troubleshooting. The second account was of a random person here in the office. So the fact it updated using this second persons account confirmed other accounts CAN update apps downloaded with a different primary account.
Now they want me to do a full restore of my device. To me this isn't troubleshooting, it's just getting rid of me and putting me on to a different support agent. I'll try it tonight anyway just so it can be ruled out.
As per others in this thread, doing a full factory reset and setting up everything from scratch isn't an option. Especially not on multiple devices. Not when I am of the opinion this problem is due to my account and will recur as soon as one of the newly downloaded apps gets another update.