It's just a set expression, Maque: hold between finger and thumb. You could say She held it carefully between her finger and her thumb ... but that's just 'cluttering' the sentence with unnecessary possessive adjectives: it's obvious that you're talking about her finger and her thumb, so you don't need to mention it.
It's not a mistake or a set expression, and you are not stupid but observant. I have never noticed this before, but we do tend to omit determiners when they might crowd a sentence.
I think this phenomenon occurs mostly with parts of something just mentioned in compounds like "between ... and ...", "(from) ... to ...", adverbial "... or ...". Some such combinations have become fixed expressions:
between her finger and her thumb
between her finger and thumb
between finger and thumb
on her finger
(from) head to toe/foot, (from) ear to ear, (from) side to side, (from) top to bottom, (from) end to end, (from) stem to stern, between street and stars, for man or beast
... lived here fifty years, man and boy
"From chair to window" would be right in a context in which the chair and the window are part of something just mentioned or implied:
The extent to which one's feet get cold when sitting in the kitchen is a function of the distance from chair to window.
In the original sentence, the thumb and finger (presumably the index finger) are part of the person mentioned as she. It might also be used for the thumb and finger of what she was holding:
She held the hand of the wounded ape between thumb and finger.
The construction "works" in this sentence, but the sentence is ambiguous. I would say the construction itself does not depend on being unambiguous, but this is not a good sentence without more context to clarify which meaning is intended.