It is indeed an pencil, not a fountain pen, and it uses refillable or replaceable lead.
Young Indy's pencil is a period-correct hexagonal metal body pencil with classical or classicist barrel ornamentation, for looks as much as for better holding. During the first two decades of the 20th century, ornamentation was very expressive, and the more expensive the pen or pricey the marque was, the more exuberant the styling became.
The pencil has a round cap and does not feature a pendant ring to attach it to a chain or cord. Instead, it already features a clasp, which you can just about make out on the DVD intro image on the left, next to the compass.
Archaeologists preferred metal pencils for fieldwork, especially basic sketching and site mapping. The British Museum as well as the PEF and the Petrie (all in the UK) have a small collection of them in storage.
They are not rare ? it's an everyday item ? and can be purchased at reasonable antiques prices today. There's a complete Astoria pencil/pen set on eBay right now, with gorgeous florentine ornamentation, which is similar to but a tad more fancy than Young Indy's pencil. It is offered for a rather ambitious price, though. But hey, someone may bite?
https://www.ebay.com/itm/ASTORIA-MO...KNIFE-PENCIL-SET-W-BOX-1910-20s-/371834089994
Best, Archaeos (back from the dead like this thread)