One of the larger branches of a tree.
One of the jointed appendages of an animal, such as an arm, leg, wing, or flipper, used for locomotion or grasping.
An extension or a projecting part, as of a building or mountain range.
One that is considered to be an extension, member, or representative of a larger body or group.
Informal. An impish child.
tr.v. limbed, limb·ing, limbs
(out) on a limb Informal
In a difficult, awkward, or vulnerable position.
If you look at the link above you will see that there is only one vague reference to its possible use to refer to "sensory or sexual organs", viz:
1 : one of the projecting paired appendages (as an arm, wing, fin, or parapodium) of an animal body made up of diverse tissues (as epithelium, muscle, and bone) derived from two or more germ layers and concerned especially with movement and grasping ***but sometimes modified into sensory or sexual organs***; especially : a human leg or arm
..so it could also refer to a nose or an ear, in that case - but by far the usual reference is to arms or legs, which is clearly what is being referred to if the context is medical.
If they meant the other kind of "member" (Glied as opposed to Gliedmaßen) they would call it by its (Latin) name - doctors don't mess about with euphemisms.