“Climate“ does not mean what it used to mean:
The Greek word that we trace climate from was klima, which means “inclination,” “slope,” or “latitude” and klima can be traced further still, to the Greek klinein, “to lean”. There was a theory in antiquity that the world could be divided into seven distinct zones called climates, which were designated based on the slope or inclination of the northern celestial pole changing as one moved north from the equator. Climate was in use in English for well over a hundred years before we began to use the word in the 16th century to refer to weather conditions.
Once climate was applied to weather, it did not take very long before we began to employ the word in a figurative fashion. From the middle of the 17th century on we have considerable evidence of people using climate as a synonym of atmosphere (in the non-literal sense of that word).
Yes, climate and incline are related and Greek. But climb, which sounds kind of like incline, and one can imagine “climbing an incline”, is just a coincidence. It’s Germanic / Old English, related to cling and clench and clutch.