Right now Kurdistan is primarily a geographical term for areas where Kurds live (and have lived for many centuries), although there are movements for political independence.
As to why they never formed a country - it comes down to bad luck.

Nation states became popular as a concept in the 19th century. As the empires of the late 19th to early 20th were falling apart that’s when current nation-states were formed, by people who could pull it off.
Kurds couldn’t due to timing and location. Just as the Ottoman Empire which ruled them previously was falling apart, a new empire, the British, jumped in to fill the gap. Kurds didn’t have time or resources to establish a state in one of the rare moments when it was possible.

In the end Ottoman Empire was reformed into Turkey, and the territory was split between them and a British “protectorate” which will eventually become Iraq.

So the Ottoman Empire was a thing and for a very long time. THere were a lot of subcultures within the empire, mostly divided along various the various Islamic factions, they were cultural regions with long standing informal boundaries, that group has historically been over there, this group has been over here, that other group has always been in that other place, we all believe some different things but we don’t need to interact with them.

Then WW1 happens, everyone gets involved with the fighting because of a huge cascade of treaties pulling more and more nations into the conflict, this included the Ottomans, who have a lot of oil reserves, which is needed for mechanized warfare. England and France join up and agree to take down the Ottoman Empire and carve it up so they have control over the oil and so the remnants of the empire don’t rise up. So they break the Ottomans, drew up lines on a map, split up and pushed together various cultural regions into unstable messes.

Those borders mostly stick around to the present day with a few more destabilizing changes after WW2, were basically dealing with the problems of a region with highly sought after natural resources, that was purposely made culturally unstable to exploit those resources.

I’m no historian or expert on the region, but my guess would be, that political boundaries are frequently somewhat arbitrary. Just look at the diversity of languages spoken in India. Cultural identities and similarities frequently cross borders.

Another example would be northeastern Thai (Isan) people in many ways has more culturally in common with Laos than central Thailand.

I mean that they don’t just go back to ww2 or anything but there has been mention of a kurdish state/country since the middle ages.