The UART is useful in next to no time. You wouldn’t want the UART to call an interrupt while the software serial is working.

It depends what you mean by "at the same time”. You could rarely receive from both software serial and UART simultaneously,
but they could definitely both work together on a chip yes.

For a UART with a chip that has some other timing interrupt running.. no problem if you wrote the thing on the other end.
So long as you leave enough duration between transmitted bytes.
The interrupt during PBP software serial command would distort a frame by extending the duration of some bits.
The only way the UART will stuff up is if you don’t check it’s receive buffer often enough, since the hardware is responsible for shifting bits,
it’s also a lot quicker to access. You only have to check a byte is present and copy it which should take PBP a few instructions.