If you were changing the 9sp to 10sp, then how does it help to share the 11sp trainer cassette with the other one on the trainer?

Anyhow purely focusing on changing the said 9sp bike to 10sp, you definitely can fit a 10sp cassette on the 9sp freehub body (some may need a 1mm spacer if I am not mistaken, but they come supplied with it). But the then again, you would want a 10sp shifter. Then depending on groupset a corresponding road rear derailleur would not match it in terms of pull ratios, for instance the latest 10sp Tiagra is different, you would need to double check if your 10sp shifter at least has same pull ratios as the 9sp one had so that the actual derailleur can stay.

You should change chain to a 10sp one as well. The rest about chainrings still applies.

Components wise the change would be cheaper, but what is the objective here? I would reconsider this because you are essentially adding only one speed and making all the spares more expensive and quicker to wear out - 10sp cogs and chains are narrower.

Since you called yourself beginner as to the bike components, I just want to emphasize all indexed shifters are for specific number of speeds. The shifter that has 9 indexed positions cannot work with 10sp cassette. The cogs are spaced differently, so do not expect that it would eg shift 9 of them correctly, not even if the pull ratios are same for the derailleurs.

You may find more detailed info to these e.g. here: https://bike.bikegremlin.com/1278/bi...compatibility/