Trumps flat out wrong about this immunity claim, and it would pretty much destroy all checks and balances if allowed, but he’s only claiming it for acts while he was still in office. Jan 6th happened while he was still legally the President. He’s claiming that he has immunity for actions as President.
Obama hasn’t been president in 7 years. Anything he does now is as a normal citizen. An equal comparison would be if Obama were charged today for something that happened in 2016, and he claimed immunity.
It would mean that Biden would be immune from prosecution for anything though now, which would set up for him just ignoring any election, legally.