Elements other than carbon that can form many different compounds by bonding to themselves?
Is carbon the only element that can do this?


No, carbon is not the only element with such characteristics.


  If not, then what are the other elements can also do this? 


There is a whole number of elements such as silicon, arsenic, germanium.


  Is there a term to describe such elements?


At least I'm unaware of such a term, which might be furnished by our far wiser community.


  What is the chemical characteristic that allows this to occur?


Catenation.

More information:

According to the Molecular Orbital Theory, the condition for a compound to exist is that it should have more electrons in the bonding orbitals than in the anti-bonding orbitals. So, as long as you have the bonding orbitals filled more, you can have pretty anything, more than just chains of atoms.

Thus, the existence of a compound also depends on the precise conditions in which the compound is kept, for example sodium forms different types of chlorides under different conditions and that as pointed out by Poutnik in the comments, $\ce{He2^1+}$ and a ton of others are discovered and still more awaiting discovery.