If you were making a small polypeptide you'd have a protecting group on the side-chain of glutamic acid, and preferably also an activating group on the carboxyl where you want to attach it. If you're just mixing glutamate with another amino acid you'll get a mixture as you expect (including all kinds of possible oligomers).
Protein synthesis in the ribosome actually works the other way around (the amine-end is in the ribosome, tRNAs are attached to residues at the carboxyl side). So your expected error would be the continuation of protein synthesis at another alpha amine: the lysine sidechain. Apparently the ribosome is quite good at distinguishing this 4-carbon difference.
My 2 minute hypothesis is that this is one of the reasons why there are residues with shorter acid sidechains (glutamate and aspartate) while a shorter version of lysine is not used in biological protein synthesis.
How the tRNA synthases are so specific that they don't load the amino acid with the wrong carboxyl group is a different question, but tRNAs are extremely specific enzymes in many ways (they have no problem recognizing the difference between a methyl and a hydroxyl group).