Is there a standard for water bottles?
Product recommendations are generally not the thing here. There is however a reputed 4-letter Swiss company that has been producing internally glazed aluminium water bottles especially for cycling, properly fitting in standard bottle cages for many decades now.

And since it is Swiss it will certainly go on for much longer. The problem however with internally glazed metal bottles is their relative fragility. cycling bottles tend to be dropped and hard contact with the ground damages the coating which makes the bottles expensive garbage.

Most modern plastic bottles no longer have that specific 'plasticky taste' that cycling bottles used to have and can be used without restriction. Also all companies produce recyclable ones and some are even biodegradable after some months when lost in the outdoors, if this helps to lighten your conscience.

Concerning cages: cages that have a certain flexibility and that are slightly tighter keep a much better hold of bottles on rough terrain. When pros ride on cobbles the team mechanics often stick strips of skateboard tape to the cages to give the bottles a better hold.