Confirming Jim Humphreys' and Marty's answers,
page 17 of (the English translation of) [Vinberg, È. B., Commutative homogeneous spaces and co-isotropic symplectic actions, MR1845642] contains:

*In the simplest case of the two-dimensional sphere $S^2= SO(3)/SO(2)$ this result [the fact that the $SO(3)$-module $\mathbb{C}[S^2]$ is multiplicity-free] was
known much earlier thanks to the Laplace spherical functions used in mathematical
physics. This is the origin of the term “spherical homogeneous spaces”.*

Note that if $X$ is an affine $G$-variety (with $G$ a complex reductive group), then $X$ has a dense orbit for a Borel subgroup $B \subset G$ if and only if $\mathbb{C}[X]$ is a multiplicity-free $G$-module.