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Mystery watchmaking tool - video
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My guess is that it is for removing tiny clock gears/spacers/bearings. The various width slots accommodate different diameter shafts. As the plunger screws down on the end of the shaft the tool pulls up on the gear/bearing/etc gently slipping it off the shaft.
My SWAG is based on the fact that I made a tool for similar purposes some time ago.
https://www.homemadetools.net/forum/...d-puller-27409
It's not exactly the same design but the function is similar.
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Just a press for pins in gears...