Hello, runner
I don't know if this will be very easy to understand. It will depend, perhaps, on how familiar you are with syntactic analisis.
"a person or object" is a noun phrase which has two heads: person and object. Both heads appear in coordination (linked by the conjunction "or").
The indefinite article "a" doesn't modify the noun "person" only; it modifies the whole construction "person or object".
"a" is a premodifier of the whole noun phrase and, within that construction are the two heads (nouns).
The construction "a person or an object", which you suggested, is also possible. Both are grammatically correct.
Miriam