Not sure if there’s anything more to the question than that. These nails are not simply in proximity to the fingers/toes and held in place by their roots, they’re clearly attached in some manner. If you press something in between the nails and the fingers/toes, the attachment breaks for a while but eventually grows back.
Skin?
Seems to me that “skin” is a good answer: Nail (anatomy) - Wikipedia
The nail probably doesn’t re-adhere, but the lose part is pushed forward and replaced by the growing nail behind it.
Is the skin sticky? (And if it is, then why won’t it re-adhere?)
The out layer of all your skin is dead tissue, the nails are just a different sort of dead tissue: “Several layers of dead, compacted cells cause the nail to be strong but flexible.”
It isn’t sticky in the sense, will adhere to new stuff placed next to it, but in the sense, sticks to the stuff next to it which has been part of the same tissue since it was formed. If you tear off the top dead layer of your skin it won’t stick back on either, but it normally stays on for the same reason.
The nail isn’t sticky, but not only is the root longer than it looks (and it goes pretty deep), the skin right underneath it is not as dead as uncovered skin; it’s like the skin inside a freshly-obtained blister (how do I know? Brother’s childhood accident). While that isn’t “sticky as glue” and as naita said, it has the general ability to stick to similar cells that gets the body’s soft parts to, well, stick together.
Short answer: the nail bed.
Slightly longer answer: People think the nail grows from a “root” and is somehow adhered to the finger by some non-nail substance. While the proximal portion of the bed does create the majority of the keratin in a nail, the distal part, the part you see past the cuticle, is still alive, and still contributes keratin to the nail plate.
To put it another way, think of your entire nail bed as one big long flat root. When you separate your nail from your finger, it doesn’t stick back down. It just grows out.
If you injure your nail bed sufficiently, you can cause problems as the scar interrupts the smooth growth of the nail. The distal portion forms a rough keratotic patch, while the proximal portion loses adhesion and is no long in continuity with the fingertip, causing it to lift or curl in a dystrophic fashion. This is a very annoying problem and sometimes requires surgery to fix.