They should be: “I don’t have AS many books as you.” and “I don’t have AS many books as you do.” They mean the same exactly the same thing. There is no difference in meaning. Both would be heard in normal spoken speech. You might see more of the first one in written English, though I can't guarantee that.

Just to let you know the difference between "so many" and "as many", I will give you two examples of sentences: 1 - I have so many readings to finish before my exam. 2 - I wish I had as many nice clothes as you do.

The final "do" in the second example is implied in the first example. I'd also suggest saying “I don’t have *as* many books as you.” Other than that they sound perfect to me.

I see! I actually didn't know that!!! Thank you for your explanation again! Best regards.

Congratulations! I see that you have learned that some people feel that in a negative sentence you should use "so": I am not so tall as you. They feel that it expresses the negative more strongly than "I am not as tall as you." But, as the other posters told you, most native speakers use "as."