No. The name change effect specifically applies when the monster is being used for a Fusion Summon. The name isn't changed at any other point during the turn.
I was wondering, since many people use Monstermorph to easily summon Gandora, the same should work to play Blue-Eyes Shining Dragon.
The question is, does playing Blue-Eyes Shining Dragon guarantee your victory in the current meta ?
Considering that Monstermorph will allow you to play up to 2 Blue-Eyes Shining Dragons in a standard duel.
If you use the equip spell Synchro Boost on a blue eyes (any of the BE dragons) to increase its level to 9 you can then use your skill to turn the blue eyes dragon into a level 10 Shining Dragon in 1 turn.
But Shining won't be protected from effects like Judgement Dragon.