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The only way to get that information is from your delivery advisor, or sales advisor (likely a service advisor can see it too, but I am not sure). What you are asking the tesla people for is the cars "born on date". That isnt printed anywhere on the car, only month and year is, but the tesla internal system will tell them the day it was "born".
I am not sure if that is date the car actually finished manufacturing, or date the main body was assigned a vin, or what (depends on manufacturing process). The born on date is as close as you will get to what you are asking, though.
I know my car was born on 11/16/2018, for example. I ordered on 11/28/2018, and took delivery on 12/08/2018. I was matched with a car that was completed already, last year when I decided to take the plunge.
If you are not in front of your car, you could also look at the window sticker. It shows the month/year of the date of manufacture. No specific date though