There's certainly a homotopy-theoretic analogue. A universal cover of a connected space X is (up to homotopy) a simply connected space X' and a map X' -> X which is an isomorphism on πn for n >= 2. We could next ask for a 2-connected cover X'' of X': a space X'' with πkX'' = 0 for k <= 2 and a map X'' -> X' which is an isomorphism on πn for n >= 3. The homotopy fiber of such a map will have a single nonzero homotopy group, in dimension 1--it will be a K(π2X, 1). (For the universal cover the fiber was the discrete space π1X = K(π1X, 0).)
An example is the Hopf fibration K(Z, 1) = S1 -> S3 -> S2.
Geometrically it's harder to see what's going on with the 2-connected cover than with the universal cover, because fibrations with fiber of the form K(G, 1) are harder to describe than fibrations with discrete fibers (covering spaces).