How can this system not be asymptotically stable?

First of all, $e \to 0$ does not in general imply that $\dot e \to 0$ (see the warning about this on p. 122).

But I don't think that's the main issue with your argument. The problem seems to be that you're going to the limit $t \to \infty$ in some terms, while keeping $t$ in other terms. But it's the same $t$ everywhere, so if $t \to \infty$ in one place, it has to do so everywhere.

One can imagine a scenario where $w(t) \to 0$ as $t \to \infty$, and where $(e,\theta)\to (0,\theta_0)$ for some constant $\theta_0 \neq 0$ (in such a way that $\dot e \to 0$ and $\dot \theta \to 0$). I haven't thought deeply about showing that this is really possible, but at least it is consistent with the ODEs: in $\dot e = - e + \theta \, w(t)$ all three terms tend to zero, and similarly in $\dot \theta = -e \, w(t)$ both terms tend to zero.

It is a standard problem in adaptive control and estimation. To ensure that $\theta$ converges you have to assume something about $w$. The required condition is known as persistency of excitation: there exist $T>0$ and $a>0$ such that for all $t$ $$\int_{t}^{t+T}w(s)w^\top(s)ds \ge aI.$$

For example, $w(t)\equiv 0$ or $w(t) = e^{-t}$ are not persistently exciting.