This kind (moving pieces) wasn't normally used to stuff meat in a hopper Mart. One piece articles (looking like a darning tool) were used to form pies, stuff meats into grinders (and were called plungers or mashers), etc.
Normally the wooden sausage stuffers were two piece and had a tapered end so that the sausage casing could fit around it while the sausage was being inserted. But this one sure resembles a sausage stuffer - especially with the lip on the outside! OR it could have been used as a primitive wine corker! However, most had an outside opening on the side - but people were multifunctional!
Can you please tell us what the diameter of the outside end is? The height of the piece? the diameter of the plunger?
If the center plunger piece was larger it would have been a early spice grinder/pounder/pulverizer! Used much in the way a mortar and pestle was used. OR if it had grooves for an attachement on the end of the plunger could have been a cookie press! Too tall for a butter press.