Is a married man's proposal to another woman, without divorcing his wife, considered an injustice to his wife, and how can one deal with society's disapproving view of second marriages?

Polygyny is not oppression because it is a divine decree from Allah. Oppression arises from husbands' misapplication of it. A husband who is not physically or financially capable of polygyny or of being just falls into oppression. No consideration should be given to any custom or tradition that prohibits what Allah has made lawful. If the young man is serious, let him propose to you through your guardian. You must cut off all contact with him completely until the legitimate marriage contract is concluded, because he is a stranger to you.