She grew up to be as beautiful a woman as her mother.
Literal
She [topic-は] grew-up [and-て] mother [with-と] same [like-ような] beautiful woman became.
同じような means 'similar to' or 'the same kind of' — combining 同じ (same) with ような (like). This variant of the comparison is slightly softer than 同じように ('in exactly the same way'), leaving room for individual variation while acknowledging the resemblance.