The word is indeed old-fashioned, from the 16th century when it meant paying court to a woman with intention of marriage.

Where does courting originate from?

courtship (n.) 1570s, behavior of a courtier, from court (n.) + -ship. Meaning the wooing of a woman, attention paid by a man to a woman with intention of winning her affection and ultimately her consent to marriage is from 1590s.