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"Topological Properties" is a generally used term to refer to properties of a topological space which are preserved by homeomorphism. Can one use the term "metric properties" to refer to the non-topological properties of boundedness, Cauchy convergence, etc which are preserved by isometry ?

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I don't believe there is a universally agreed upon name for what you're describing. It's worth noting that there are two other common types of homeomorphism lying between isometries and homeomorphisms. We have the following hierarchy.

  1. Homeomorphisms - bijective maps continuous in both directions
  2. Uniform homeomorphisms - bijective maps uniformly continuous in both directions
  3. Bi-Lipschitz homeomorphisms - bijective maps Lipschitz continuous in both directions
  4. Isometries - bijective maps that preserve distance

Each of these preserve different things. Homeomorphisms preserve topological properties, and uniform homeomorphisms preserve uniform properties such as completeness, sequences being Cauchy, and total boundedness. Isometries clearly preserve everything, so that leaves only bi-Lipschitz homeomorphisms. We could call properties preserved by these "metric properties" or "bi-Lipschitz" properties. The main property preserved here and not by uniform homeomorphisms that comes to mind is boundedness.

My impression is that most people would understand what you meant if you said "metric properties" or "non-topological metric properties."

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    Thanks. I'm curious to know what would be a property that is preserved by isometry but not by a bilipschitz homeomorphism ? – 2017-01-24
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    Well, it's a bit of a cheeky answer, but: distances themselves. Just because $d_1(x,y)$ and $d_2(x,y)$ are comparable for all $x,y$ doesn't mean the geometry induced by the metrics is "the same." One can see this quite clearly if $X$ is a normed vector space and $T: X \to X$ is a linear homeomorphism (i.e. a homeomorphism that's linear in both directions). Then in fact $T$ is a bi-Lipschitz homeomorphism (continuity of $T$ and $T^{-1}$ are equivalent to boundedness as linear maps). The distances $\Vert x\Vert$ and $\Vert Tx \Vert$ may give rise to very different geometries. – 2017-01-24