2006 Ion manual shifter
 

My 2006 Saturn Ion 5-speed shifter is sticking. I can shift when the car is turned off, but once I start the car, it won't shift.

That would be a clutch issue.

I am puzzled as to how it is a clutch issue. I put it in gear (1st)
, start the engine, let out the clutch and it grabs in the same position, moving the car.

If the clutch does not fully release you will not be able to shift the transmission with the engine running.

Wouldn't the fact that I can be going down the road, push in the clutch and take it out of gear mean that the clutch is working?

If the clutch is not completely releasing, the synchronizer's in the transmission cannot work properly. If you can pull it out of gear and not shift it into the next gear try pumping the clutch.

No.Take your foot off the gas and shift without touching the clutch and it'll shift.

BTW You have a clutch problem.