As long as you can touch the target you can cast shocking grasp on it
Since shocking grasp has a range of touch, your bugbear must be able to reach the target before the melee spell attack commences. The rules don't explicitly define how far a creature can actually reach their arms and touch something. However, your trait is called "long-limbed" and it allows you to make melee attacks from 10 feet away, so it is incredibly reasonable to assume that you can reach that same distance for the purpose of simply touching something.
So, talk to your DM and make sure they agree that it is reasonable to allow you to use your long arms to touch the target of the spell. If they agree, everything works like you want it to.
However, to clarify the general confusion about melee spell attacks:
Melee spell attacks are melee attacks
Melee attacks are a category that include melee weapon attacks and melee spell attacks. Thus, the feature applies to melee spell attacks as well.
A melee attack typically uses a handheld weapon such as a sword, a warhammer, or an axe. A typical monster makes a melee attack when it strikes with its claws, horns, teeth, tentacles, or other body part. A few spells also involve making a melee attack. (PHB 195)
And Jeremy Crawford has clarified and agrees on Twitter:
A melee spell attack is, indeed, a melee attack...