In logic, "not (A or B)" is the same as "(not A) and (not B)"
Similarly, "not (A and B)" is the same as "(not A) or (not B)"

You can apply these De Morgan laws to English sentences.

But first, the meaning of the sentence has to be clear.

"I don't like A and B" can have several different meanings:
"(I don't like A) and (I don't like B)"
"I don't like C" (where C is the combination of A and B)

If the first meaning is the intended meaning, then you might argue that these statements are the same:
"I don't like A and B"
"(I don't like A) and (I don't like B)"
"not (I like A) and not (I like B)"
"not (I like A or I like B)"
"not (I like A or B)"
"I don't like A or B"

But notice how the first and the last statement are different? Where did exactly did this transformation take place?

The answer is that the first and the last sentences have multiple meanings.

However, these more precise statements are the same:
"(I don't like A) and (I don't like B)"
"I don't like (A or B)"

Whereas these two are different:
"(I don't like A) and (I don't like B)"
"(I don't like A) or (I don't like B)"

This is the heart of the problem. English is ambiguous. The intended meaning can usually be inferred by a native speaker. But a non-native speaker might associate a different logical meaning with the same sentence. That's where the confusion lies.

But there are other, smaller complications as well:

It's not necessarily true that "I don't like A" is the same as "not (I like A)." You may be indifferent to A. In that case, "I don't like A" is false, but "not (I like A)" is true. So in reality, the two statements are not necessarily the same.

The reason for this is the third value--indifference--in addition to like and dislike: This makes it a trinary logic system, rather than the usual true or false binary logic system. In real life, there may be logic systems like this.

Another complication is that the English phrase "A or B" can mean "A or B or both." But the same English phrase can also mean "either A, or B, but not both."

For example:
"Clean your room or go to bed now!"

The "or" in that sentence has a different meaning than the "or" in this sentence:
"I would like some ice cream or chocolate."

Another consideration is that sometimes the same English words can be used to describe things that are different. For example, someone may like chocolate and ice cream (A and B) when it's ice cream with a little chocolate swirled on top, but would NOT like chocolate and ice cream (A and B) when it's essentially a bowl of chocolate with just a spoonful of ice cream on top.

If there is a distinction between these two cases, then the statement "A and B" is not precise enough. It needs to be clarified further in order to understand what was intended.

English speakers usually able to infer the intended meaning. But it can happen that the same words can be interpreted differently, and when this happens, it is a major source of confusion. Many "paradoxes" are based on this sort of thing.