Time: Does time have physical properties, or is it just a measurement?

Just a measurement used to describe from point A to point B in regards to existence. It is a measurement because there are different forms of measuring "how long it's been:" minutes, hours, seconds, year, month, etc.. Existence has physical properties and time measures how long it has existed.

Our mathematical representation of physics in spacetime uses something called a tensor. Conceptually a tensor describes an objective, coordinate system independent ‘reality’. In other words, Ma Nature doesn’t give a hoot how we humans chose to measure space and time, or any physics in spacetime

But, we humans like to be able to measure things with respect to space and time. So, we define coordinate systems (reference frames) in order to do that. When we do that, we assign coordinate system dependent meanings to the components of tensors. In this case, time and space become properties of measurements rather than relating to an objective reality.

It appears to me that when time is evaluated with carbon dating, that time has an atomic value.

It is just a measurement. Earth going around the Sun was one unit called a year. Earlier people used to use water clock, sand clock etc. Now they use the energy transition in a atom as the basic unit

0ne grows with time physically

It is one of the basic structures of the universe, along with space.  Without time, space as we know it would not exist.  So, space (meaning everything you consider "physical"), rests on there being time.  You can't touch time, but it's there.