My dad made this rabbet plane too. It's for cutting rebates in wood. Rabbet/Rebate? I can't find any explanation but you have to wonder about that name? A dialectic thing? Or just because it's got that sticky-up bit and reminds you of a rabbit. Well, actually, it doesn't really does it? If it wasn't for the name, you wouldn't think, "looks like a rabbit that does" would you? I've had a dig around but no one on the internet seems to want to talk about it.
I suppose it doesn't matter really. What matters is that this beautiful piece of routed hardwood, with it's decades old construction marks and wear was used to carve grooves and recesses into machined wood. Say for, oh I don't know, perhaps for a glazing bar where it makes provision for the insertion of the pane of glass or to accommodate the edge of a cabinet's back panel or for a casement window jamb or for shiplap planking.
Oh, the word "rabbet" is from the Old French "rabbat" meaning "a recess into a wall".