Yes, "ought" is the past tense of "owe". At some point, the second alternative spelling "owed" was introduced to better separate the two meanings (literal and figurative), but it's still the same word; a similar thing happened with "flower" and "flour", those used to be interchangeable spellings of the same word but then somebody decided that the two meanings of that word should be separated and given specific spellings.

And the construct "you owe it to <person> to <verb>" still exists even today but is not nearly as popular as "you should <verb>" precisely because it has to state to whom exactly your owe the duty; with "should" it sounds like an impersonal, quasi-objective statement of fact which suits the manipulative uses much better.

The etymology makes a connection through old English. Oxford dictionary also contains this meaning:

> used to indicate duty or correctness

A duty to others is something you owe them; think, a duty of care and its lack, which is negligence.