Which is grammatically correct: "woke up by the..." or "woke up to the..."?

The first sentence is not what I would consider normal, idiomatic, native-speaker English and should be this:

Sophia was awakened by the rattling sound of her washing machine.

The second sentence is grammatically correct and standard English.

Except in an extremely contrived context, I don't think OP's first example is valid. Google Books returns just 205 instances of woke by the sound, and it seems to me almost all of those have the verb form wrong. They should be woken by the sound, which itself returns 15,300 hits.

You're woken by something if that thing is what causes you to wake.

You wake to something if that's what you first perceive (usually, hear) when you wake up.

Note that although the by form is normally used in the context of unplanned awakenings, it can be used of a deliberate method - I made sure I woke [up in time] by using a loud alarm clock.

Actually they both seem grammatically correct. 1st one seems to say that she actually woke up "due to the" rattling sound and the 2nd one seems to suggest that she woke up by herself but heard the rattling sound going on.

Yes, both are grammatically correct; it really depends on what you want to express. If you want to say that she woke up because of the rattling, then use the first one. If you want to say that she woke up on her own accord, then use the second one. However, having said that, the first one would be better stated if you said, she woke up "from" the rattling sound.