There are the allusions to marriage, of course, of course I see that.
The oath being the promise of marriage that without it, the end must come.
But there is, too, the literal thing of Ophelia making an end to her song.
And, la, la, la, she keeps getting interrupted by this pretty talk.
I’m sure she’s heard oaths like this for much of her young life…maybe she’s just asking to be able to finish her song in peace. It’s a reasonable enough request.