Leonardo da Vinci became renowned for his multiple talents: he was a painter, architect, engineer, mathematician and inventor.

Your example works because:

• you can use and then:
  • In speech, elide all conjunctions, up to (and arguably, including) the last;
  • In writing, replace all but the last conjunction with commas

• 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:

  • Singular
  • Indefinite (a/an rather than the)

  Not that it is either a or an, but just the above information

• and because conjunctions can coordinate - join - units at any level - and the result is the same unit:
  • Noun and/or Noun forms another "Noun"
  • Noun Phrase and/or Noun Phrase forms another "Noun Phrase"

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.