If she were the princess, she would sing all day in the castle. Why use she were?

"were" is used because the verb (to be) is in subjunctive mood (虛擬語氣), which expresses a condition which is doubtful or not factual.

If she were the princess, she would sing all day in the castle.
Apparently, "she", as mentioned in the sentence, is not a princess, so subjunctive mood of the verb ("to be") must be used.

"Subjunctive" is most often found in a clause beginning with the word "if". It is also found in clauses following a verb (e.g. ask, demand, insist, order, suggest, wish, and etc.) that expresses a doubt, a wish, regret, request, demand, or proposal.

Subjunctive - form of the verb:
1. verb to be = "were" (past tense or non-factual in if-clause) or base form "be"
2. other verbs = base form of the verbs