Is there a name for the class of groups in the title, and any sort of characterization? Free groups and surface groups are in the class, but presumably there are many more...



Groups where every two generator subgroup is free
There is no name, there are lots of examples. Guba gave many non-trivial examples in Guba, V. S. Conditions under which 2-generated subgroups in small cancellation groups are free.
Izv. Vyssh. Uchebn. Zaved. Mat. 1986, no. 7, 12–19 and here: Guba, V. S.
A finitely generated simple group with free 2-generated subgroups. 
Sibirsk. Mat. Zh. 27 (1986), no. 5, 50–67. There are of course infinitely generated locally free non-free groups. Every proper ascending HNN extension of a free group is an extension of such a group by a cyclic group. See also Arzhantseva, G. N. Olʹshanskiĭ, A. Yu.
Generality of the class of groups in which subgroups with a lesser number of generators are free. Mat. Zametki 59 (1996), no. 4, 489--496, 638; translation in 
Math. Notes 59 (1996), no. 3-4, 350–355. 


