Imho it might be the only time in your career when you are truly, 100% your own boss. You own your PhD thesis. While your professor/advisors give input and feedback you ultimately decide what goes in there and what not. You have the freedom to explore anything you deem as relevant and there are no customers/co-workers/VCs etc who tell you otherwise.
My limited world view saw that nearly all grad students had to work on the projects that their PI got funding for. Their own research was never financially possible unless they got significant funding from outside their lab. Even my SO, who got grad school fully paid for by a smaller government agency, couldn't work on individual research and had to do what the PI said.
I don't think those words mean what you think they mean.
A more accurate description would be that you are your advisor's slave.