Is an updated assumption of command required after Commander is promoted to a new rank but still the company commander or is it still valid?

The commander stays in command until a new commander is appointed. It depends on the division, brigade and/or battalion commander's policy. In 2003 we had an O-6 battalion commander because the brigade commander (now GEN Milley) had a policy of no battalion change of commands while being deployed. We also had a Major as a company commander because his company was a cross fill from another MACOM. All that said there is a responsibility when possible the higher command needs replace a company commander upon being promotion to Major. This sometime takes time I stayed in command 4 months till a new commander was available.

It's not nessesary, the person and signature is the same.

No. Unless he just wants the assumption of command orders changed. The delegation of authority form with signatures only changed when delegation changes