Meaningful words and exchange of vows are just that. Marriage, on the other hand, means: the legally or formally recognised union of two people as partners in a personal relationship (historically and in some jurisdictions specifically a union between a man and a woman).
Just to help clarify, the below is from the interview transcript where they both lie:
Meghan: Right? Like this kind of stuff. It’s so, it’s so basic, but it’s really fulfilling. Just getting back down to basics. I was thinking about it — even at our wedding, you know, three days before our wedding, we got married . . .
Oprah: Ah!
Meghan: No one knows that. But we called the Archbishop, and we just said, ‘Look, this thing, this spectacle is for the world, but we want our union between us’. So, like, the vows that we have framed in our room are just the two of us in our backyard with the Archbishop of Canterbury, and that was the piece that . . .
Harry: Just the three of us.
Oprah: Really?
Harry: Just the three of us.
Meghan: Just the three of us.