Is the Nikon D750 a professional model?

I know the term professional is over used, but in terms of capability, performance, and build quality is the D750 considered a pro camera?

There are no "pro" cameras. There are pros who use cameras.

Any camera can be used to take photos by a professional, but no, that camera is not considered one of Nikon's professional level cameras.

The D750 is a full frame FX camera. Most pros use a full frame camera.
I use cropped sensor DX cameras. And I can get professional results with them.
The skills of the photographer are more important.

The term "professional" cameras would indicate that the camera is very robust and has a shutter that has been stress tested beyond 200.000 shutter cycles. It will also have a metal chassis and environmental seals to keep dust and moisture out of the camera. All the "pro" level Nikon cameras also have a built-in interval timer and 51 focusing points available.

I have a colleague who uses the D750 as a backup camera to his Nikon D3s and D810. He shoots a whole lot of sports with those cameras.

If you visit the DXOmark website, you will see that they consider this camera to be of "professional" level in its quality

Nikon does not claim the D750 to be one of their "professional" models, but it's built in such a way, and has a good host of features/capabilities that many professionals use it.

I wouldn't get hung up on this issue since there are loads of photographers using all sorts of cameras to achieve a desired look or a specific result. In short, if the camera does what you need it to do and it allows you to express your personal vision, then use it. Regardless if it's a D4s, D750 or a Holga, the gear really doesn't matter just as long as it doesn't impede on your ability to express your vision.

A pro could make good photos with that camera, as he/she could with many cameras.