You will need to open up the saw to find out for sure.
The most common cause of wobble like this is that the bearings or bushings are worn out. These are the things that hold the main shaft in place. They should be a close fit to the shaft, and the shaft rotates within them. When they are worn out, there is too much room and the shaft flops around or wobbles.
In the linked parts diagram, these would be items 13, 814 or 818, and possibly 39. I notice that two bearing sleeves are listed on the side of the page under "popular parts"; no surprise there.
When you open it up, you will likely find that one or both of the bearing sleeves is not fitting nice and smooth on the shaft. It will probably look like an oval or egg-shaped hole in the bearing instead of a perfect circle.
I recently did this replacement on an electric hand drill. Same situation-- bushing was worn, drill chuck wobbled too much, replaced the bushing, now it runs smoothly.
(As to the name bearing versus bushing: if it has an inner and outer ring with balls or rollers in between, it's usually called a bearing. If it is a single part where the shaft just rotates directly in it, it's usually called a bushing. But both are doing the job of holding a rotating shaft. If they want to call it a bearing sleeve when someone else might call it a bushing... well, it's still doing the same job so no big deal)