"No quiero pizza porque no tengo hambre."
Translation:I don't want pizza because I'm not hungry.
No, in Spanish "Tener" is used, also for being thirsty, cold, hot, and other bodily conditions. Please read this article:https://www.thoughtco.com/how-to-use-tener-3079782.
Essentially, you are saying: "I do not have hunger" It is using the verb tener (to have) as an idiomatic expression. You would have come across this when referring to your age. "Tengo doce años" I am 12 years old (Or direct translation word for word, I have 12 years)