Are professional association membership fees usually paid by one's institution? An academic usually is a member of a professional association. E.g., a professor in meteorology can be a member of American Association of Meteorology. Sometimes, the fee can be something like $300 per year. Is it always the case, that such membership fees are covered by the University (department) or a research institute? Are there exceptions?

The goal is to understand the prevailing norm (e.g., 40% (or 60? or 80?) of universities usually cover single association membership up to $500/year).
As you've already seen from the comments, there is no single solution to this problem. The rules vary greatly from country to country, institution to institution, and even department to department. You will have to check with the specific rules for your funding agencies, department, and other "revenue sources" to determine precisely which sources of funding can be used to pay for these. 

(For instance, in Germany, a professor's "discretionary budget" could definitely be used to pay for such things; in the US, this may or may not be the case.)