Is marriage still relevant in today's society or is it an outdated concept?

Marriage doesn't go out of style, it's a divine institution.

Do people in a relationship need to get married and if so, why?

Yes, because marriage is a legal, moral and sentimental bond that sustains a couple's relationship.

Should people marry for tax benefits, moral reasons, religious customs, etc.?

People must marry for love, to share their life with another being who is their moral, religious and emotional complement.

Or is it enough to be an engaged couple and there's no need for that piece of paper?

It is not enough, the role is an important symbol is a contract that is made with the other person.