Some consider both words in both... and conjunctions:
The conjunctions should be carefully positioned and their conjoined elements should be well balanced. That is, what follows both and what follows and should have the same grammatical form.
whereas others differentiate them:
Both is paired with and to add emphasis to two coordinated elements in a sentence. Both is a focusing adverb; and is a coordinator. Together, they are a "correlative conjunction".
Chambers seems to agree with the latter:
both adj, pronoun (sometimes both of something) the two; the one and the other
- She burnt both hands on the stove. (adj.)
- I'd like you both to help. (pron.)
adverb, as well; both something and something not only one specified thing but also another specified thing
- He is both rude and ignorant.
What is sure, is that both... and is not a determiner, as it does not only correlate nouns or pronouns, but also verbs, adjectives, adverbs, even prepositions:
Both… and… can link not just verbs but also nouns, adjectives, adverbs and prepositions. (guinlist.wordpress)
The same site specifies the real role of both in this correlative conjunction:
BOTH cigarettes AND cigars are harmful to health.
This statement is equally about cigarettes and cigars... The presence of both gives an early indication that a list of two items is being given.
That's not to say that both cannot be a determiner. If you look it up in Cambridge, you will see that it needs to be followed by plural nouns or pronouns, if it is to function as a determiner:
We use both to refer to two things or people together:
- Both those chairs are occupied, I’m afraid. (The two chairs are occupied.)
- Are both your parents going to Chile? (Are your mother and father going to Chile?)
Also, you can see contexts where both... and both... is possible, where both is used as a determiner:
Both front doors and both front wheels of my car were damaged in the accident.
So, since both is inherently plural, it cannot determine a singular noun as in the sentence you indicated as not possible. You cannot say:
*both Jack and both Jane
but you could probably say
both Jacks and both Janes (four people in total).