Hello Evgeniia. It’s possible to say it either way, although much more common without “got” in this particular context.
Explanation:
First of all, we can ignore “every day” — the object is “class”, “I have class.”
Your dictionary is correct: "We use have (got) to talk about possession, relationships, characteristics and illnesses."
When we say that we “have five lessons”, the lessons are not something that we possess, and they are also not a relationship, a characteristic, or an illness. “To have lessons” is an expression meaning “to attend class” or “to study with a tutor”, or something like that. There is no possession. As another example, the idiomatic expression “to throw a party” means the same thing as to “have a party”, but we cannot say “to have got a party”.