Taking the question literally -- no. If you take a Fender P or a Stingray, and sand the name off the headstock, the tone won't change.
Tone is the result of design, materials and construction. A Fender P sound like a Fender P because of the way it was designed, the materials its made of, and the nature and quality of the construction. If you go to a good luthier and ask him or her to make you a nice copy using the same materials, it will sound the same. If the luthier puts more time or care in that the Fender factory, then it may even sound better.
There is always, though, a little bit of black box in tone. As every piece of wood is different, sometimes the particular blocks of wood used on a particular instrument will just sound better than the blocks of wood used in a different instrument. In those cases, you can't tell until the instrument is made and you play it.
And then the effect of playing regularly over several also shakes the fibres in the timber down, and somehow the instrument sounds better as a result.
The last two factors can make a considerable difference. I have a Mexican Fender J that I've owned for 15 years that sounds better than some new US-made Fender Js in the store.