No, you can use your US E-upgrade bank. No problem. The assumption is that people don't have dual airline status when they write up the procedure.
I did it once. Also, I wound up with US and UA miles for the flight.
An complicating scenario though is when a UA preferred is regular status on US and tries to upgrade a non-Y/B fare using the US e-upgrade bank. (For example, someone loses status on US by focusing on UA but has some e-upgrades left in the bank.)