Have a look at /dopusdata\ConfigFiles\colorgroups.oxc on both computers.
Each label will be defined by a line in that file, and each has a unique ID which is automatically generated if you create a new label (and predefined for the stock labels).
Any labels that you want to use on more than one machine will need to have the same ID on both. The names that you see (e.g. "Red") don't actually matter, only the IDs (e.g. "{2F499E2B-8B6E-4B80-8E3B-C73E032E801F}").
Preferences / Favorites and Recent / Labels / Automatically store labels in the file system if possible and its parent option will also need to be on.