I was wondering what a beacon's beam is or what it's made of. (e. g. Is it an entity etc.) and how I can have a beacon beam without a beacon.
Beacon beams are specially rendered… things, which don't fall into any category like "entity" or "particle", just like lightning, fishing rod lines and leashes.
You cannot have a beacon beam without a beacon. Not even having a beacon tile entity without a beacon block works, it just gets deleted. The closest you can get to it is retexturing the beacon block to be invisible.