Yes, Kurds are the most likely descendants of the ancient Medes, but also other ancient peoples. Kurds are a heterogenous nation, which means there isn't just one ancient predecessor, but multiple ones. Kurds are therefore not wrong to claim descendant from multiple ancient nations because it makes perfect sense. In fact, there are millions of people who identify as Kurdish in the ethnic sense, even though they are linguistically farther removed from the main Kurdish languages (like the Zaza Kurds). The Kurdish ethnicity is really complex, so is their history. For example, "Kurdish" in its own is a group of languages instead of just 1 language. And if you travel through Kurdish lands, you'll notice big differences not only linguistically, but also religiously and physically.

The Kurdish language is, just like the Median language, a North-Western Iranian language. In addition, the area where the Medes were dominant and founded their empire are predominantly Kurdish nowadays. Furthermore, as far as the High Middle-Ages, "Medes" and "Kurds" were used to refer to the exact same people by different nations. E.g. the Armenians called Kurds in Kermanshah region as "the Medes", whereas Arabs called the exact same people "Kurds". Around the Middle-Ages, for some (to me unknown) reason, "Kurd" slowly replaced "Mede", but it also replaced other (tribal) names and slowly became an ethnic and cultural word. E.g. the Mitanni, Milan, Alan, etc. were all proto-Kurdish clans, and these names still exists as "old tribal names" among Kurds.