It’s complicated, and Congress is taking advantage of that to keep him there. The nine-member United States Postal Service Board of Governors has sole authority to fire, and hire, the Postmaster General. At the moment there are only 7 members, 3 Dems, 3 Reps, 1 Ind. 6 members are required for a quorum.
Biden has 2 pending nominations and a pending reappointment of an existing governor. There are still 2 governors remaining from Trump’s administration at this point. The Senate confirmed 3 seats in July 2021 and 2 seats in May 2022, and nothing since. They seem to be doing the absolute bare minimum to intentionally delay any possible changes.
Note that these are not full time government positions like the Postmaster. They only meet once a month usually.