Haha those are great questions-- My best guess is: We call him Joshu and Joshu was the name given to him by people who believe in identities, but did he call himself that? As for not thinking ill will, I think that in Buddhism even "thought" itself is an illusion. So encouraging someone to think well or not think poorly of others isn't Zen either.
I am no zen monk so I really can't answer your questions satisfactorily, but the book "Dropping Ashes on the Buddha" explains zen buddhism to start.
https://groveatlantic.com/book/dropping-ashes-on-the-buddha/
"In a cookie factory, different cookies are baked in the shape of animals, cars, people, and airplanes. They all have different names and forms, but they are all made from the same dough, and they all taste the same.
In the same way, all things in the universe–the sun, the moon, the stars, mountains, rivers, people, and so forth–have different names and forms, but they are all made from the same substance [...]
Their names and their forms are different, but their substance is the same. Names and forms are made by your thinking. If you are not thinking and have no attachment to name and form, then all substance is one. Your don’t-know mind cuts off all thinking. This is your substance. The substance of this Zen stick and your own substance are the same. You are this stick; this stick is you."