My chair suddenly started lurching precariously from side to side, and when I took a closer look I noticed this inner ring around the cylinder that keeps the chair centered had popped out and vomited a bunch of grease everywhere.
I can push this thing back in and it'll sort of click into place but a few minutes of sitting later the chair starts wobbling again.
Is the chair base itself warped or damaged? The place where the cylinder sits?
IF the base itself/the hole that the cylinder sits in IS bent, warped, damaged then unless you're okay replacing both the cylinder and the base at a cost of probably $80-$90 which is getting pretty close to the cost of some budget chairs then you may have to consider whether it's better to just get a new chair.
Usually the base itself is just the 'arms' that hold the wheels plus a central hole that the hydraulic cylinder fits inside snugly (you often have to hammer it to get it out again when replacing but it can be done).
If the base is fine however you can buy a new piston which should fit 90% of office chairs (no idea about gamer chairs) online that comes with a toolkit for doing the replacement for $40 on bezos-net.
If this is not a normal office chair and I'm not sure it is from the picture (it should have levers under it for adjusting height) then I'm less sure.
I'm pretty sure the base is okay. It is, however, a budget chair- a budget gamer chair. I have some old broken office chairs that I've kept around (the backs and arm rests fell off because they were cheap plastic, the seats and bases are fine. I'll check if the pistons are interchangeable
They do sell the tool-kit for removing the pistons by itself on the aforementioned website and probably other places. It's basically two ring collars that snap/screw together with a screw and hex key to turn it to push the pieces apart though that may be the easy part as the pistons tend to be pressed into the base and getting that out can be a trouble and require some bashing with the thing braced and in some cases just may not come out without using a method that destroys the piston (not a problem when replacing it but a big one when trying to swap).
The one in your picture may be a cheaper design than what I'm thinking of here. I'm not a chair expert I just once replaced a cylinder on an office chair I salvaged and followed the instructions and watched a couple videos.