Pager system charging pin and socket
They appear to be what is know by the brand name "pogo-pins", which I believe are officially called "spring contact pin".

Quite a few manufacturers do them. They tend to be expensive and have a low current rating compared to other contacts, but are useful in that they are so versatile, as only exposed metal is all that's required as a mating half. Often small ones are used on on test pads of PCBs to do production line tests.

EDIT: just noticed you ask to find the mating half in your question. This type of contact often just contact to a bit of metal in some custom designed housing. But they are also used with a proper contacter, which as a profile to match the pin (in this instance the pin has sharp spike, so the official mating half would have suitable cone indent in it).