This is by a trivial dimension count.
As I suspect this is missed due to vague definitions, here is a short summary (which is not necessarily canonical for contemporary research, even though it is faithful to the original definition by Anosov, as far as I'm concerned):
If $M$ is a $C^1$ manifold (not necessarily compact) and $\mathfrak{g}$ is a $C^0$ Riemannian metric on $M$, then a $C^1$ flow $\phi_\bullet: \mathbb{R}\to \operatorname{Diff}^1(M)$ (not necessarily volume preserving) with generator $X$ (i.e. a nowhere-zero vector field on $M$ such that $\forall t\in\mathbb{R},\forall p\in M: X(\phi_t(p))=\frac{d}{dt} \phi_t(p)$) is $\mathfrak{g}$-Anosov if there is a continuous $\phi_\bullet$-invariant splitting $TM = S(\phi_\bullet) \oplus \langle X\rangle \oplus U(\phi_\bullet)$, where $\langle X \rangle$ is the rank $1$ subbundle of $TM\to M$ generated by $X$; $S(\phi_\bullet)$ is a nontrivial subbundle consisting of vectors contracting exponentially fast w/r/t/ $\mathfrak{g}$ under the action of $\phi_\bullet$ and $U(\phi_\bullet)$ is a nontrivial subbundle consisting of vector expanding exponentially fast w/r/t/ $\mathfrak{g}$ under the action of $\phi_\bullet$. $\langle X\rangle$ is the center bundle of $\phi_\bullet$, $S(\phi_\bullet)$ is the (strong-)stable bundle of $\phi_\bullet$ and finally $U(\phi_\bullet)$ is the (strong-)unstable bundle of $\phi_\bullet$. By definition both $S(\phi_\bullet)$ and $U(\phi_\bullet)$ have rank at least $1$, meaning that $TM$ has rank at least $3$, i.e. $\dim(M)\geq 3$. Thus the case $U(\phi_\bullet)_x=E^u_x=0$ is discarded by definition.
(When $M$ is compact, the dependency on $\mathfrak{g}$ can be dropped: if $M$ is compact and $\phi_\bullet$ is $\mathfrak{g}_0$-Anosov for some $C^0$ $\mathfrak{g}_0$, then it is $\mathfrak{g}$-Anosov for any $C^0$ $\mathfrak{g}$.)
Anosov originally called Anosov flows (on closed manifolds) (U)-flows (alternatively $(\Upsilon)$-flows). Here is the relevant part from his monograph Geodesic Flows On Closed Riemannian Manifolds With Negative Curvature (p. 6):
Just to tie in with the paper you cited, a $\mathfrak{g}$-Anosov flow $\phi_\bullet$ is codimension 1 if at least one of $S(\phi_\bullet)$, $U(\phi_\bullet)$ is rank $1$.
Finally let me mention that there are generalizations to trivial stable or unstable bundles (partial hyperbolicity, beyond-uniform hyperbolicity, ...) that cover the case you were considering; the Anosov case is very strict and structured compared to these.