You could export both to JSON, move the data over manually, then import the JSON data into a Spine project. It may not be too hard to select the entire animations section and replace the animations section of the other file. This can fail for a number of reasons, eg if an animation keys attachment visibility or deform, that attachment must exist in both projects. Also all the keyed bones, slots, etc must be the same in both projects.
Spine has an Import Project
feature which can bring skeletons from one project into another:
Import - Spine User Guide: Project
Spine allows bones, slots, and attachments to be dragged from one skeleton to another in the same project, but you currently can't drag skins or animations, so this may not be helpful for you.
However, Import Project
can also import an animation from another project, though currently it can only import one animation at a time. Like when editing JSON to add animations to a skeleton, this can also fail if the skeletons don't have the keyed items, but you likely will get friendlier error messages than when editing and importing JSON data.