[...] Leonardo da Vinci became renowned for his multiple talents: he was a painter, architect, engineer, mathematician and inventor.
In the above sentence, why has only one "a" been used, even for words that begin with a vowel?
Leonardo da Vinci became renowned for his multiple talents: he was a painter, architect, engineer, mathematician and inventor.
Your example works because:
When you join units at the Noun level, the Determiner only needs to "agree" with the first in the series, as it's arguably for phonotactic reasons - what sequences of sounds are permissible in English, which the writing system has come to reflect.
The more abstract, implicit "Determiner" can simply have the same meaning of the explicit Determiner, the:
Not that it is either a or an, but just the above information
In your example, what is happening is something like (let's ignore the final comma, for the purposes of this demonstration:
Det = Determiner Cnj = Conjunction Noun = Noun  = Elided/replaced Conjunction Leonardo da Vinci... a painter and architect and engineer and mathematician and inventor. Det Noun Cnj Noun Cnj Noun Cnj Noun Cnj Noun becomes... Leonardo da Vinci... a painter, architect, engineer, mathematician and inventor. Det Noun Noun Noun Noun Cnj Noun
Another possible valid conjunction would be, as you point out one that has Determiners for each Noun (forming a Determiner Phrase (more commonly known as a Noun Phrase), or Nominal Group):
NP = Neterminer Phrase DP -> Det + Noun Leonardo da Vinci... a painter, an architect, an engineer, a mathematician and Det Noun Det Noun Det Noun Det Noun NP  NP  NP  NP Cnj an inventor. Det Noun NP
You might need to be a bit imaginative with the above interlinear explanation. The plain text is the first line, and the second line, as with the above examples, is the word class of each constituent - I'm analysing the commas as realising a grammatical function here. The third line is a higher-level of categorisation.
Now, if you're technically inclined, my understanding is that generative grammar doesn't allow for a node to have two parents, and so you couldn't have paratactic (equal) relationships between each of the above noun phrases or nouns. You would be forced to have a Noun Phrase formed by two coordinated Noun Phrases, which are themselves formed of two coordinated Noun Phrases... if you have Noun (Phrases) which are not a power of 2 in number, then you're forced to place one above another. I'll gladly edit if I've misunderstood.
A simpler example is: 'I eat with [a knife] and [a fork]' > 'I eat with a [knife and fork]' (which emphasises that I use the knife and fork together) compared to 'I eat with [a bowl] and [a spoon] (which emphasises that those are two different kinds of utensils' > ?'I eat with a [bowl and spoon]'.
Saying 'he was [a painter], [an architect], [an engineer], [a mathematician] and [an inventor]' might emphasise that he was all of those things separately, but part of his genius was that he was often mixtures of those things, or even all of those things (and more) at the same time.