Quake's engines are open source. Its levels, textures and music are not. That distinction is the whole answer.
id Software has open-sourced the source code for Quake, Quake II and Quake III. That is why you see browser builds at addresses like q1, q2 and q3.
Open source code is not the same as a free game. The engines are free to reuse, but the game content, meaning the maps, textures, sounds and music, remained commercial. Those browser builds either use the small freely released demo content or expect the real files from a copy you own.
For the later games, id released the full source under the GPL, which is why Quake III derivatives are all over the place, but the asset rule never changed.