Do nails with points on both ends exist?

Yes, but they're far from common and can be difficult to use. Picture two nails with the heads "welded" together with the points pointing directly away (180 degrees away) from each other. That's a double-ended nail. The problem is setting the second nail shank. The first shank (end) of the nail is driven into whatever is being fastened by using a hollow driver that slides over the shank of the outside half. Just drive it in. But to set the second side of the nail, you have to beat on the wood or whatever it is you're nailing together. This can damage the wood, and can be a real hassle in a lot of (or most) applications. The biscuit joiner systems are all probably a better choice for joining wood rather than a double-ended nail. And the adhesives (wood glues) used today are fabulously strong (when applied correctly to properly prepared surfaces). There are some folks who have patented double-ended nail systems, but why bother? They're just a pain for most applications. Think about how you, as the user of the fasteners, are going to go about setting the second half of the nail to construct, say, a picture frame. It may be possible to jig the frame up and actually press the thing together to set the fasteners, but gosh, how tedious. Good luck with that.