The role of the tension in the linear string is to create a force that will pull back on any deviation from a straight line. This is what causes waves to propagate along the string. A point is moved away from equilibrium, and the tension acts to get it back to its equilibrium position. Things don't stop as the string now has kinetic energy, and keeps on moving against tension.
In a bulk material, there is already a force that will try to reduce any deviation from equilibrium. You don't need the tension, and the equation for a string is not valid. As you guessed, that force related to the elastic properties of the material. One thing that should not be confused is that in solids, you can have longitudinal waves and transverse waves, which will propagate at different speeds. There are different properties of the materials (Young's modulus, shear modulus, bulk modulus and Poisson's ratio) that must be used for each specific case. The propagation speed will also depend on the density of the material.
The physics in the 1st approximation are worked out here:
https://en.wikipedia.org/wiki/String_vibration
with the result that frequency depends on the length, tension, and mass density as:
$ f = \frac{1}{2L}\sqrt{\frac{T}{\mu}} $
The 1st factor is why bass strings are long and treble are not (on a piano). The lower 3 strings on your guitar have a tension producing string wrapped in wire (dead weight) so that you can increase $\mu$ while still maintaining a reasonable tension (to low: it sounds bad, to high: snap goes the guitar).
Of course, strings on instruments are not infinitely thin--and their finite width shifts the harmonics up from their canonical frequencies (inharmonicity). The n-th frequency is:
$ f_n = (nf_1)\times [1+(n^2-1)\frac{r^4\kappa}{TL^2}] $
were r is the string radius and $\kappa$ a material modulus (with units of pressure).
Longer strings are less affected by this, so concert grands are really long.
See the Railsback curve to see how this affects piano tuning in surprising ways. (For a physicist, it as fascinating "fail" of the 1st approximation).
Best Answer
The classical string equation that you are referring to, is formulated by making a number of assumptions, which include that the vibration of the string does not affect its tension. This makes Young's modulus irrelevant for results calculated from the idealized equation.
In the real world, materials with low moduli of elasticity will follow the ideal equation more closely, since the tensions will change less during vibration. For materials with a higher modulus of elasticity, I would expect that: