As an American who was raised in the Mid-Atlantic, I could see using class and lesson(s) interchangeably, but I would never use class without the article unless there were multiple students AND multiple lessons/sessions and even then I might still use the article.
- I haveHe has (a) piano class every Tuesday.
- I haveHe has piano lessons every Tuesday.
- I haveHe has a piano class/lesson today.
Oddly, I think that if I said "I have“He has piano classes every Tuesday” some people would think I hadmeant multiple, separate classes on each Tuesday.