Learn Turkish: Memur kimliğimi görebilir mi?

It uses the ability structure -(y)abil/-(y)ebil. The verb root is gör- (see), followed by -ebil, then the aorist tense ending -ir. Finally, the third-person question particle mi is written separately.

I know the word for ID is kimlik. Why is it spelled kimliğimi here?

Two things are happening. First, you add the possessive suffix -im to mean "my". Because it starts with a vowel, the k at the end of kimlik softens to a ğ, making it kimliğim. Then, you add the accusative case suffix -i to the very end.

Why do we need that second -i at the very end of kimliğimi?

That is the accusative case, which marks a specific direct object. You are not asking if they can see just any ID; you are asking if they can see your specific ID. Since it is the direct target of the verb görebilir, it needs this marker.

If I am talking to the officer, why don't I use the "you" form (görebilir misiniz)?

In this sentence, memur (the officer) is a third-person subject. You are asking someone else: "Can the officer see my ID?" If you were speaking directly to the officer, you would usually drop the word memur and ask: Kimliğimi görebilir misiniz? (Can you see my ID?).