Your sentence structure is: Someone gave (to) Somebody else Something as Role.
Consider the following:
- Alex gave Arnold some worms as bait.
- Bob gave Barry a degree as Principal.
In the first instance, the worm (not Alex) is the bait. In the second, Bob (not the degree) is the principal.
Context matters a great deal in these kinds of sentences.
In your example, there are a few problems with taking “customer service” as “the driver making deliveries”. First, “customer service” is typically a static office that customers access, not the delivery person. Second, if “customer service” operating as “the driver” was giving a third party the phone number of a fourth party, conformity to language norms is probably not their top concern.
Now, there are some issues with treating “your number” as “the driver making deliveries” as well, most notably that numbers typically don’t drive or make deliveries. However, if we take this as metonymy, or (more likely) just an imprecise identification, the sentence as a whole communicates the intent well.
Conclusion: your example asserts that the speaker was speaking to the person that customer service identified as the driver, and that customer service provided the speaker with the driver’s contact information.