Follow XY
Adds controls for following only the X or Y position of another layer. Useful when you need to track on one axis. To use (to track one axis) have two copies of the layer, do the track normally on one and apply it (stabilize). Now select the other layer and apply the preset on the other layer. Select the tracked layer as the source and toggle the follow checkbox you want.
Installing
Download follow xy.ffx and put it in %USERPROFILE%\Documents\Adobe\After Effects 2022\User Presets. Make sure to put it in the AE versions folder you actually use. Doesn't work in AE versions under 2022.
Now the template can be found from the menu Animation > Apply Animation Preset...
Alternatives
You could also right click the position on both layers and choose "Separate dimensions" and then use the 
pickwhip to link just the X or Y position to the other layer. This template just creates easy to toggle controls for the same thing.