No. There are two parts to the legendary rule. One is written explicitly in the official rulebook:
If a second copy of it would come under your control, instead the second copy is discarded.
This part handles any cases where you would be able to trigger the “Arrives:” ability of a Legendary unit while already controlling another copy of a unit with the same name. It says that if you try to do that, the new copy doesn’t arrive at all.
The other part is written a bit less explicitly:
You can never, ever have two copies of a given legendary unit at the same time, even if some other rules implies you can. This supersedes all other rules about legendaries.
This tells us some facts about what kind of game states we expect to see, but it doesn’t tell us e.g. which copy of DeGrey to get rid of if a player controls two copies of DeGrey after a Polymorph: Squirrel expires. In my rules engine I think I am keeping the one that has been DeGrey under the controller’s control the longest, or the one that has been DeGrey the longest if there is a tie for that, or the one that is oldest if there is a tie for that, or a random one if there is a tie for that.