The name is not the player's name as you can see, and it is displayed in the log as “Rabbit” as well.
Once you become a monster, there will be a “monster skill” option added to your action command menu. (Apologies that the below is only in Japanese, but the highlighted option is “monster skill”)
From this option you can select and execute abilities, which will require and consume TP.
Currently the rabbit’s level is pretty low, so I only have Foot Kick at my disposal, which is why I am out here fighting this mandragora. Once I level up I’ll start learning Whirl Claws and other abilities.
We wanted to become a really sweet monster and show off how you can fight against other players, but the PvP battle system is still being worked on, so we were unable to do it at this time.

It’s also possible to upgrade (evolve?) to a Black Rabbit or a White Rabbit. Depending on what kind you become the abilities will differ. For example, the White Rabbit will learn Snow Cloud.
While I am sure a great deal of players are in fact looking forward to having a new PvP aspect added to the game, those who do not fancy PvP much will still be able to enjoy this content (i.e. changing into a variety of different monsters). However, since part of a monster’s daily activity is to battle with adventurers, there is a high possibility that we will not be making it so you can completely avoid PvP on the monster-side.