There's one in Tempe, AZ that has a 1980s "Dialed digit recorder", but that's a much later microprocessor controlled unit.
The mechanism is that of a "Morse inker", a recorder used to receive telegraph signals. Here's one that's part of a Siemens telegraph set from 1862 or so.[1] The wiretapping version is the receive part of a telegraph set made portable in a nice wooden box with a carrying handle.
All this shows how rare wiretapping once was. Law enforcement had to use repurposed antique telegraph equipment.
Any idea of museums that may hold historical pen registers (or paraphernalia) for public viewing?