Brand and Fizz's passives are linked to active skills, and thus get the benefit of ability-based items (same with jax's ult passive).
Teemo's passive is purely passive and thus can't even proc Liandries.
lol jk, the real reason is because this game has 0 consistency and every ability works differently.
Gangplanks Q counts as a basic attack, so it will proc on hit effects but not ability effects, even though it can be spellshielded. Warwicks ult also does on hit effects but will not proc sheen-based items.
Jax's W will proc spellvamp, Nasus' Q will proc Lifesteal. Poppy's Q turns the entire attack (All it's AD too) into a magic attack, yet still procs on-hit effects (possibly in addition to ability effects, just like Jax's W)
If blitz is blinded, his knockup will miss, if Xin is blinded, his knockup (and pretty sure all 3 Q attacks) will hit.
Some on-hit abilities work on towers, some do not.
Some passive stack abilities (such as caitlyn counting up headshots) will count up on towers, some will not.
The list goes on.