Having surgery to correct the configuration of the physical body to match the mental image applies to both conditions but that does not mean the conditions are related.
For example, you can use aspirin as a pain killer or a blood thinner. The fact that you might use the same treatment (aspirin) doesn't logically imply that the underlying condition being treated are related.
Similarly, even if there is a parallel between surgically changing the configuration of the body between BIID and GID, that doesn't causally relate the two conditions.
The whole body map thing has nothing to do with gender identity. It's the physical representation of the body configuration within the somatosensory cortex where this "map" resides. Strictly speaking this is theory but Ramachandran used this theory to demonstrate that, post-amputation, this map gets redrawn which explains (a) why touching the face of one patient post-amputation led to the sensation of being touched on the now-phantom limb and (b) a method for the possible treatment of conditions around phantom limbs). So in the case of amputees with phantom limbs and quite possibly for BIID, there is a mismatch between the physical configuration of the body and the somatosensory homunculus.
In the case of gender dysphoria, the situation is somewhat different. The somatosensory homunculus is not mis-mapped to the physical body, it matches exactly - the problem is that both of these conflict with gender identity because the anatomical gender doesn't conform with the mental gender.
Really, one has to look far beyond the superficial similarities to see that GID and BIID are fundamentally different things.