Could you please advise how to resolve the discrepancy in displayed emails between the PC and mobile versions of Thunderbird when using a single-login POP account?

The solution is to set up accounts in IMAP. This allows both phone and pc to always see all the messages.

"Specifically, there is a discrepancy between the Android and PC versions regarding which accounts can be logged into and which accounts can receive emails."

The problem with sharing a POP account is that each device will download and delete from server in default configuration. That means the message you see on phone is not available on PC and the message you see on PC is not available on phone. If you are experiencing something unusual, please share the setup on phone and on PC and describe the discrepancy. Thank you