If anybody can tell me what this type of box is called, it would be much appreciated.
Good question, having a random go at it - plastic numismatic storage box, <however many slots for coins it has>
This one has slots to slide in the coin holders. It work for you? Same company also sells the coin holders.
Anything having to do with money, search using numismatic as one of the terms, it refers to money as a collectable.
Heh i’ve been trying all night. I’ve found a ton of coin boxes, none of them like that. The link is for a type of box that requires the coins to be in holders first, but thanks though. I haven’t found a box anywhere yet where the coins are stored without holders, stacked, and in a cheap plastic container. The closest so far holds coins flat in indentations, but it won’t come close to the space efficiency of this one. I’ve emailed the seller too, hoping for a reply.
Try here:
Sure-Safe coin shippers
Here’s one for silver dollar.s seems like its a match
Looks like that’s the one, thanks. 
Too bad I need one like that for quarters. 