LR mobile can only sync with one catalog, so you can't use it to sync between two computers, whether they are Windows, Mac, or a mix.
That said, there are solutions for sharing a single catalog and images across two computers.
Here's a video I recorded describing my process:
any chance using Creative Cloud Files?
It's clunky, but if two catalogs refer to the same images, you can pass some settings back and forwards between these two catalogs via external metadata on the file. These images may be literally shared, or else separately stored but with changes physically transferred (replicated / synced). But LR Mobile is AFAICT no help at all in this.
Collection membership cannot be replicated this way - to do that directly, you would instead need some method based on passing Catalog data back and forth, as mentioned in another reply. However, as a workaround, if you were to apply a characteristic keyword, you could make a Smart Collection in each place which was based on the presence of that keyword. This alters the problem, into getting the same keyword loaded into images within each of two Catalogs - and that CAN be done via external metadata. However, AFAIK via this workaround, this grouping of the images will not then show up in LR Mobile in the same way, in the form of a standard Collection. However, the keyword should be visible to LR Mobile in due course.
Caveat: besides being clunky, the above's a little hazardous too:
- Whenever you write metadata to file, LR goes ahead and unselectively obliterates everything that was previously there in the file. It won't warn you, if this operation overwrites older edits onto newer ones.
- Whenever you read metadata from file, LR goes ahead and unselectively updates all that image's settings in the active Catalog. Again, it won't warn you, if this operation overwrites older edits onto newer ones.