"She not only sings but she also plays."
"She not only sings but also plays."
These carry the same meaning (with the 'play' as new, stressed information): she does both. The second 'she' is OK and common, but not necessary, as it is redundant.
''She plays the piano as well as she sings."
"She plays the piano as well as sings."
These are different in meaning. The first tells us that she has equal skills in both; the second merely tells us she does both (again with 'play' as new, stressed information). In this pair of sentences, the second 'she' is critical to meaning.