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...
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.
There is a widely used term binary finite (бинарно конечная, in Russian) group, see papers of Shunkov, Chernikov and their school. This means a group where all 2-generated subgruops are finite.
For instance, the Kourovka Notebook contains the following question (still open, as far as I know).
Question 4.74(b) (V.P.Shunkov, 1973). Does there exist a simple infinite binary finite 2-group?
Also, I stumbled across binary solvable and binary nilpotent groups... As you can guess, I am hinting that you may call your groups binary free if you like this terminology.