The indefinite article a applies to both thing and person. If you have a list, or just two items, joined by and or or (possibly other things, can't think of it off the top of my head), a lot of things that go before them apply to both/all of those items.
I painted the walls and I painted the fence.
Gosh, isn't that unwieldy? Well, we can simplify it:
I painted the walls and the fence.
In this case, the verb applies to both of those as objects equally. In just the same way:
I painted the walls and fence
The the is covering both walls and fence. Similarly:
Do you own a cat or do you own a dog?
This can become:
Do you own a cat or a dog?
And even further:
Do you own a cat or dog?
Adjectives can do it too:
I see lots of blue cars and bicycles.
That's a nice tidy sentence, but we could write the equivalent longer:
I see lots of blue cars and blue bicycles.
And longer still:
I see lots of blue cars and lots of blue bicycles.
And yet longer:
I see lots of blue cars and I see lots of blue bicycles.
The shorter version is preferable in many cases, and the longest versions are utterly unnatural.