The black plastic "pad" under the neck plate is a common sign of a fake/partscaster, though it was not an unpopular addition in the 90s. A lot of Asian manufacturers did use them.
Fender *NEVER* used that on any US or Mexican production.
A Custom Shop would have had a visible serial number somewhere, either on the neck plate, or the headstock.
That pickguard is not correct for a '62/63... note the top center hole and compare the position to the one you linked. That is a late '63+ pickguard design.
I'm not sure exactly what that is. Early Telecasters had their serial number on the bridge plate, and I think they just said "Fender" on the neck plate, but that plate looks too new.
The edges of the body look to be too sharp.
This is what the back of a legit CS would look like:
View attachment 438940
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