Is the young man considered a legal husband to the woman after he asked her to swear on the Qur’an to accept, and then announced the presence of witnesses to that, despite her separation from him?

What the man claimed about the marriage is not valid. A woman cannot marry herself, and a marriage contract is not concluded except with a guardian (wali) and two just witnesses. Furthermore, the questioner's swearing on the Quran that she accepts him as a husband is considered a promise of marriage, and she has the right to retract it whenever she wishes. In conclusion, what happened is not a legitimate contract.