I am sorry if my question is stupid (or very hard) or common knowledge, or should be placed at math.stackexchage.com. As long as a math student read the definition of Lie group, several natural questions appear instantly:

Is it true, that any compact manifold admits Lie group structure? (NO)

Is it true, that there exists compact manifold that admits different Lie group structures? (YES)

Answers to these questions can be found here - Lie Groups and Manifolds

But I was not able to find the answer for third most natural question: Which connected compact manifolds admits unique Lie group structure? (As pointed out by YCor, without the connectedness assumption, there are trivial non-uniqueness examples.)

Thanks a lot for your answers!