And he wonders, What makes a Jazz and P sound different?

Of course, I understand the pickup configuration, but do these alone make the difference in the sound? Pretty much the same wood and bridge, etc...

If you routed a Precision body with JB pups, why wouldn't it sound like a Precision? Or vice versa? Or would it?

Nope, it's pretty much comes down to type and location of the pups. Past that, no difference. I mean, we all know that other factors come into play like the wood and tolerance variations of the Electronics. But yeah, you could take a P body, route it for J pups, and it'll sound like a J.

placement is the key, you can tonally shape, but that is secondary.