What is complexity of this max-edge subgraph problem?
The generalized problem (such that each vertex $v$ has degree at most $b_v$) is called the $b$-matching problem. Chapter 31 of Schrijver's book is devoted to the subject. It can be solved in strongly polynomial time.

The case that $b_v=2$ for every vertex $v$ has been studied, particularly for providing lower bounds for the traveling salesman problem.