Your question has two main parts, what's elastic about gluten, and why don't other plants have this unicorn we call a gluten protein. I'll provide some background info first, but feel free to skip ahead to the spoilers if you like.
Quick and dirty background on proteins...
All proteins (like gluten) are made of differing sequences, and number of about 20 or so basic amino acids(glutamine, lysine, cysteine, arginine, isoleucine, just to name a few). So when we talk about large proteins like gluten, it's sometimes hard to remember that chemically, we are talking about many-many amino acid (AA) molecules which link together. The way the AA's link together matters because, the chemical interactions create the overall larger shape, and helps determine the physical properties of the final protein.
If I didn't put you to sleep, or you want to geek-out on this stuff: resources regarding the secondary(bonding between AA's which creates alpha-helices and beta-pleated sheets), tertiary (AA side chain interactions and AA's backbone), and quaternary (3D) structures of proteins, pubmed is where you need to go.
Glutens and what they are:
Gluten is the main storage form of proteins in wheat. Gluten itself is a fusion of glutenin and gliadin protein units which belong to the prolamin superfamily(prolamines are cool, look them up).
Gluten subunits fall into three broad categories:
- High molecular weight (HMW) prolamines, (glutenin subunits) approximately 6-10% of protein
- Sulfur-rich prolamins (approximately 70-80% of protein) which include: gamma-gliadins, alpha-gliadins, and the B- and C-type low molecular weight (LMW) subunits of glutenin
- Sulfur-poor prolamins (approximately 10-20% of protein) include:
v-gliadins, and D-type LMW subunits of glutenin
Variances in the spread of these proteins contribute to the way wheat varietals are utilized commercially(like winter or summer wheats...you have seen the variations available to buy on store shelves). The composition of gluten protein in whatever wheat strain that's being used translates into differing physical properties(elasticity, density, etc) in the dough it makes.
Where do the elastic properties come from?
Elastic properties in gluten come from the way the components of the protein matrix interact with water. The gluten proteins react with water in ways that other storage forms of protein cannot, namely in the swelling/hydration bit. There are two things going on at the same time. The gliadin sections of the gluten protein contribute to the friction with which the hydrating liquid is able to move around in and near the protein matrices(viscocity). The glutenin portion of the molecule is responsible for the elasticity or strength of the protein matrix. Glutenin seems to undergo reversible (not as common as you'd hope) stretching because of the way it is folded.
Why don't any other grain proteins seem to have gluten?
I would like to point out this is more of a plant physiology/phylogeny question rather than one related to cooking.
They just don't have what it takes.
The way different plants create the storage form of their proteins is determined by their specific DNA sequences within the genome. AA Structure, order(N-C terminus) and final shape of the proteins created by the plant are a result of the transcription, translation and post-translational modifications of DNA sequences in the plant.
References for those that want them:(I did try to find free online ones)
Cereal seed storage proteins: structures, properties and role in grain utilization
Wheat Gluten Functionality as a Quality Determinant in Cereal-Based Food Products
Annual Review of Food Science and Technology
Vol. 3: 469-492 (Volume publication date April 2012)
Distribution of gluten proteins in bread wheat (Triticum aestivum) grain
The formation and properties of wheat flour doughs
Wheat Protein Composition and Properties of Wheat Glutenin in Relation to Breadmaking Functionality