I had the same thing a few years ago, a little gray spot that would move in my field of vision, and it freaked me out at first. It's called "floaters" or "eye floaters" in general, and it's caused by tiny particles in the vitreous fluid of your eye. It's super common and in the vast majority of cases it's completely harmless. The thing is, if it really hasn't changed since May and you're seeing fine otherwise, that's a good sign. The floaters that move with your gaze - that's exactly because the fluid in your eye moves with your eye movements.
That said, I'm not going to tell you it's nothing you need to get checked out - you should still see an ophthalmologist to make sure there's no retinal detachment or something like that going on. It's especially if it comes with flashes of light, a sudden increase in floaters, or a shadow appearing in your peripheral vision that you really need to hurry. Since it's been stable for two months, you can book an appointment without panicking, but might as well get it checked with a pro to sleep easy. At 34 and without other vision problems, it's really probably just a harmless floater.