Yes.
Going off the address, it seems highly likely that she was Elizabeth Musk.
She may then have given the Kershaw name because the father was a Kershaw.
Then it becomes unusual if the child is the one we think was raised by ‘grandmother Ann Kershaw’, why did she marry as Musk. Presumably because she was registered as Musk.
Why was she not raised by a Musk family member, when they lived closeby.
Or, more unlikely, that an informal ‘adoption’ had been arranged prior to birth.
DNA might be the only way as you say.