Well, no, not necessarily. Being born into the caste is certainly the most common and easiest way of being samurai, but people definitely were "promoted" to being samurai if they performed some exemplary deed. Or were a foreigner a high-ranking daimyo took a liking to.

Hell, one of the most important samurai figures in all of feudal Japan, Toyotomi Hideyoshi, started life as a peasant, was promoted to samurai and then one of the most historically impactful daimyos in Japanese history, shogun in all but name.

Ironically, his birth as a peasant was the reason he couldn't become shogun (he had none of the necessary bloodlines). But samurai, that was definitely possible. Uncommon, but possible.