There's basically no way an LLM would come up with a name for itself that it consistently uses unless it's extensively referred to by that name in the training data (which is almost definitely not the case here for public data since I doubt anyone on Earth has ever referred to Grok as "MechaHitler" prior to now) or it's added in some kind of extra system prompt. The name seems very obviously intentional.
There is the original prompt, which is normally hidden as it gives you clues on how to make it do things the owners don't want.
Then there is the chain of thought/thinking/whatever you call it, where you can see what its trying to do. That is typically on display, like it is here.
so sure, the prompts are fiddled with all the time, and I'm sure there is an explicit prompt that says "use this tool to make sure you align your responses to what elon musk says" or some shit.