It’s a popular usage, and didn’t start with Biden.
When Pence is no longer vice president (assuming he doesn’t next become President) reporters will refer to him as vice-president as well.
It used to be that a person retained the title of the last highest office they held - a retired Ambassador remains Ambassador So-and-So for the rest of his life - unless/until he attains a higher position.
The exception has traditionally been for offices which are only held by one person at a time - president/vice president. But the current popular usage is that Trump will always be ‘President Trump’, and Clinton ‘President Clinton’; Biden and Pence ‘Vice President’ - unless the latter two manage to become president, in which case they will be referred to thereafter as ‘President’.
This didn’t start with Biden.
I understand that but with the current president there, calling Biden vice president was just...icky.
It’s like Trump is at a party with his wife Melania and Marla is introduced as his wife.
I think former vice president is fine but otherwise it just seems tacky.