If the only evidence for dark matter we have is gravitational, why call it 'matter' rather than 'observed space-time geometry that does not match predictions'?
General Relativity tells us that mass distorts local space-time but as far as I know, it doesn't say that it is the only thing that does so - isn't for all we know space-time is more fundamental than mass and can have local distortions caused by the big bang / inflationary epoch / flying-spaghetti-monster blueprints?
As the universe expands you would expect the density of the stuff in it to decrease for the obvious reason that the amount of stuff remains constant while the volume increases. So you'd expect something like:
$$ \rho \propto \frac{1}{V} \propto \frac{1}{a^3} $$
where $a$ is the linear scale factor (i.e. $a^3$ is the volume scale factor).
However things are more complicated than this. For radiation and highly relativistic matter the energy density decreases as $a^{-4}$, while for a cosmological constant the energy density doesn't decrease at all but remains constant. It is only matter, and specifically pressureless matter, that decreases as $a^{-3}$.
And the stuff that we call dark matter decreases as $a^{-3}$ just like regular matter, so whatever it is it behaves like matter. Hence the name dark matter.
The name, both the "dark" and the "matter" parts of it, is a historic accident.
The difference between the motion of stars in the Milky Way and that predicted by Newtonian gravity has been known since at least 1884, pre-dating General Relativity by three decades. The obvious candidate to explain this was ordinary matter such as planets, dust clouds, and burnt-out stars, hence "dark (too cold to give off light) matter".
Non-luminous baryonic matter remained the leading candidate even after Einstein developed GR (at the distances and scales involved, there isn't much difference between the Newtonian and relativistic motion predictions), so there was no real reason to change the name. It's only very recently that astronomers have found solid evidence that there isn't enough conventional matter to explain things, and with a century of usage behind it, it's going to be very hard to change the name.
According to the Einstein-Cartan version of gravity, there are two sources which "distort" local space-time:
As an alternative to missing/dark mass, we could attribute the "observed space-time geometry that does not match predictions" to dark spin current (or dubbed as " flying-spaghetti-monster", if you will).
As per Poplawski, the spin current/torsion approach has another nice property that "averts the unphysical big-bang singularity, replacing it with a cusp-like bounce at a finite minimum scale factor, before which the Universe was contracting".
