Some people think glue bottles are made from a special plastic that repels glue. This is easy to check. If you put a drop of glue on the bottle and allow it to dry, does it stick? Does it matter whether you try this on the inner or outer surface of the bottle?
Is it evaporation of the solvent, or is it something in air that makes glue sticky? It may be complicated to check every component of air, but you can easily prepare a gas that isn't air, to see what happens. For example, you could mix baking soda and vinegar or sublimate dry ice to make carbon dioxide gas. This gas is heavier than air, so you can fill a container with it. If you put a drop of glue inside the carbon dioxide to dry, the solvent will evaporate. Is that enough to make the glue sticky? Place an opened bottle of glue inside the carbon dioxide atmosphere. Does the glue stick to the bottle? Or, does the glue seem pretty much the same as before? What other processes might affect what makes glue sticky?