Does spin really have no classical analogue?

You're probably overthinking this. "Spin has no classical analogue" is usually a statement uttered in introductory QM, where we discuss how a quantum state differs from the classical idea of a point particle. In this context, the statement simply means that a classical point particle as usually imagined in Newtonian mechanics has no intrinsic angular momentum - the only component to its total angular momentum is that of its motion, i.e. $r\times p$ for $r$ its position and $p$ its linear momentum. Angular momentum of a "body" in classical physics implies that the body has an extent and a quantifiable motion rotating around its c.o.m., but it does not in quantum mechanics.

Of course there are many situations where you can construct an observable effect of "spin" on the angular momentum of something usually thought of as "classical". These are just demonstrations that spin really is a kind of angular momentum, not that spin can be classical or that the angular momentum you produced should also be called "spin".

Likewise there are classical "objects" that have intrinsic angular momentum not directly connected to the motion of objects, like the electromagnetic field, i.e. it is also not the case that classical physics does not possess the notion of intrinsic angular momentum at all.

"Spin is not classical" really is just supposed to mean "A classical Newtonian point particle possesses no comparable notion of intrinsic angular momentum". (Note that quantization is also not a particular property of spin, since ordinary angular momentum is also quantized, as seen in e.g. the azimuthal quantum number of atomic orbitals)

It's seemingly unappreciated by many people that there are different classical limits of quantum mechanics. At least there are two, a particle limit where you take $\hbar\to 0$ and $ω\to\infty$ while holding $\hbar ω$ and $n$ (particle count) fixed, and a wave limit where you take $\hbar\to 0$ and $n\to\infty$ while holding $n\hbar$ and $ω$ fixed.

In my experience, phenomena that disappear in the particle limit are often called "purely quantum" even when they survive essentially unchanged in the wave limit. Intrinsic spin is one example; the Aharonov-Bohm effect is another. Maxwell's electrodynamics should be purely quantum by this definition, so I suppose a secondary condition is that the phenomenon has to have been (re)discovered by a physicist after the 1920s, so that the claim isn't so obviously wrong.

The Dirac equation is also often called purely quantum for reasons that are unclear to me – perhaps simply because it contains a factor of $i\hbar$ in Dirac's arbitrarily chosen units. It's a classical spin-½ wave equation that just happened to be first discovered by someone who was looking for a relativistic version of Schrödinger's equation.

The meaning of spin at the classical or first-quantized wave level is described in "What is spin?" by Hans C. Ohanian (Am. J. Phys. 54 (6), June 1986; online here).

An essential difference is that there is no representation of spin in ordinary $3D$ space$^\dagger$. Unlike the spherical harmonics $r^\ell Y_{\ell m}(\theta,\varphi)$ which can be expressed in terms of spherical (and eventually Cartesian) coordinates, such a representation in terms of "physical" coordinates is not possible for spin-$1/2$ (or half-integered spin in general).

$^\dagger$ see Gatland, I.R., 2006. Integer versus half-integer angular momentum. American journal of physics, 74(3), pp.191-192.