Because Somali don't speak the language that "-stan" comes from?
It'd be like asking why we don't call German, "Germanistan".
Right, because that's Persian. Which is mostly spoken in, surprise surprise, Persia, which is notably not any of the 'stans. Which means it makes equally little sense.
More research tells me that Somaliland thinks that it is the successor to British Somaliland, which refers back to when Somalia was divided into the French Somaliland, British, Italy, etc, none of which, I feel the need to point out, are Germany. However I will concede there is more logic in that it is a British name, but damn, they are isolated as hell in their quest for independence. In that they are functionally independent but it's irrelevant because no one cares.
Persia has been the cultural epicenter for its region since Neo-Assyria and Babylon fell; it's had a wee bit of time to leave its mark on the Turkic peoples, in the Caucasus, and along the (modern) Indian hinterland. It's rather unsurprising that most of the -stans thus exist in the regions culturally tied to Persia (or the regions where these peoples migrated), from Pakistan and Rajasthan in the east, on north to Kazakhstan and Bashkortostan, over to Tatarstan, Hayastan (Armenia), Dagestan, and Kurdistan. By contrast, Arabs had their own cultural epicenter for much of this period in Arabia and Egypt; that's why you won't see nearly as many nations comprised of Arabs or those who existed within the Arabic hegemonizing influence using it. The only odd part about Somaliland is that they actually transliterated it back into Somali, but even that's not that surprising when you consider that there was no real independent, unified power that dominated the region, but rather a succession of independent minor sultanates that were subsumed by the Ottomans, Egyptians, and eventually the British. Any native name would favour individual clans, and of the non-native options, the British are the most recent and most prosperous, not to mention the most distant.