The classical objects which correspond to quantum strings have been identified, they are extremal black holes. Geons are (unstable) gravitationally bound configurations of massless fields, and they cannot be identified well with strings because they are not primitive holographic objects, as black holes are.
The physics of string world sheet, with its emission vertex operators which correspond to all the spectrum of the theory, is only sensibly interpreted as a holographic reconstruction on the world-sheet of the surrounding space-time theory. The connections between world-sheet physics and space-time physics was one of the seeds that led to AdS/CFT, and there is no longer any confusion regarding the classical notion corresponding to the world sheet--- it is an extended charged black hole.
The correspondence is more general, so that the D0 branes of IIA theory used to construct the matrix version of M-theory are point black holes, while the D3 branes are described by 4d gauge theory, and most recently, that the M2 branes by a strange 3 dimensional Chern Simons theory.