If $I^2=I$ and $I$ is finitely generated, it is generated by an idempotent $e$, and thus it is a direct summand of $R$ (since $R=eR\oplus (1-e)R$).
Since $R$ is Noetherian, all ideals are finitely generated and hence by the last sentence they are summands of $R$. Consequently $R$ is semisimple, hence Artinian.
Rings for which $I^2=I$ for all ideals are sometimes called "fully semiprime rings," and probably a more popular name I'm unaware of. Another way to summarize the above argument is that commutative fully semiprime rings are von Neumann regular and a Noetherian von Neumann regular ring is semisimple.
(Bonus unrelated discussion: I started learning about fully semiprime rings when learning about noncommutative rings whose ideals are all prime ideals. If I remember correctly, all ideals are prime iff $R$ is fully semiprime and the ideals are linearly ordered.)