It’s correct grammar. The word “much” does *not* modify the adjective, but actually modifies the adverb “more”. For example: “Much more intelligent”. The reason you’re confused is that the word “more” is replaced by the suffix -er in “better”. Likewise, we could say “much smarter”.

Both sentences are good and "much" is an intensifier as you point out. It is the case here as well. "Better" on its own is a vague word and "much better" is also vague, but is better than just "better". People don't speak this way, but if "better" means, let's say, "50% better" then "much better" could mean, let's say, "75% better". Here is an example of how the difference might matter. Suppose you have been sick and your boss phones you to ask if you are better. If you say "I'm better" that might be a way of saying "I am not ready to come back to work today, but perhaps tomorrow". If you say "I'm much better", I think it is reasonable to assume your boss would expect you back to work today. I hope that is helpful.

As you say, "much" is an intensifier and makes "better" stronger. For example, if I was sick and I feel just a little better but still sick, I would say "I'm better." If I feel completely recovered, I would say "I'm much better." Some people might tend to exaggerate, and say "much better" or "way better" when they don't really mean it. But grammatically, there is nothing wrong.