"angular momentum" is accepted because of the similarity with orbital angular momentum as explained below, and "intrinsic" means that we do not know what that "spin" actually is and where it comes from. And indeed, that's just a name, not meaning that there is really a rotation in the meaning of classical mechanics.
At the beginning, from the analogy that rotating charge generates a magnetic moment, people called that "spin angular momentum" and even imagined a real spin or rotation. That name has been accepted although they were wrong.
The reason for accepting the name, "intrinsic angular momentum", can be clarified in two side. On the one hand, spin has many same properties with orbital angular momentum, such as the commutation relation- the fundamental of quantum mechanics- between 3 spatial components, so it's "angular momentum". On the other, it's impossible for point particles to rotate in the meaning of Newton mechanics for many reasons and no experiment has discovered that rotation, so it's "intrinsic", like that the charge of a electron is "intrinsic", which often means that we do not know why it exists, where it comes from, and even what it is.
Actually, the spin can be deduced from Dirac's QM, but that's another thing. Anyway, that name, "intrinsic angular momentum", can be understood from the point view of history.