I think it depends on the gym. It usually is in reference to meet fees, coaches fees, etc.
At DDs old gym, there were no assessment fees. There was a one time buy in to the booster club and the cost varied depending on the level your DD started at the gym in.
At her new gym, there are assessment fees each competitive season. We have 2 here, Fall and Spring. Fall is the main season, assessment fees are $200. This covers the team fees, coaches pay/travel to meets, upgrades to equipment, education for coaches, etc. Spring season it is $115, only if they compete in the spring.
Those fees do not cover meets. Old gym had a booster club and we paid the actual meet fee -10% that booster club covered. New gym we pay about $85-95 per meet.
I know others pay assessment fees here as well, but they are different that how mine is. Hopefully they will chime in soon. I think it varies widely because each gym sets up their meet fee payment differently.
Another example, at old gym, had a booster club. Gym tuition was paid to gym, meet stuff, was paid to booster club. At new gym, no booster club, gym tuition paid to gym and then added on to our bill each month are the extras, leo/warmup, meet fees, assessment, etc. We usually get a breakdown of what will be due each month at the beginning, so we aren't in shock, LOL.