Is Dark Matter called "Matter" only because of gravity? As far as my understanding goes, dark matter is nothing but an amount of gravitational force, from yet unresolved/undiscovered source(s), needed to explain some observed attributes of our universe. Then, is it called "Matter" only because of gravity? Because other properties of matter like occupation of some space, mass etc is not found in this case. And, can it be some new kind of force other than gravity?
 A: As the universe expands the density of matter goes down. For example if the volume of some specific region of the universe doubles then the density of the matter in that region halves. More precisely, suppose we take the scale factor of the universe, $a(t)$, to be unity right now and we take the current average density to be $\rho_0$, then at a time $t$ the density will be:
$$ \rho(t) = \frac{\rho_0}{a^3(t)} $$
This should be intuitively obvious. Suppose the universe doubles in size, i.e. the scale factor increases from $1$ to $2$, then the volume increases by a factor of $2^3 = 8$ so the density falls to $\rho_0/8$.
But ...
Even though this seems intuitively obvious it is only true for matter, and strictly speaking it's only true for pressureless matter (though to a good approximation the matter in the universe is pressureless). For photons the density as a function of time is given by:
$$ \rho(t)_\text{photon} = \frac{\rho_{0\text{photon}}}{a^4(t)} $$
Note that this has an $a^4$ dependence not $a^3$. And for dark energy the density is independent of time:
$$ \rho(t)_\text{de} = \rho_{0\text{de}} $$
(assuming that dark energy behaves like a cosmological constant).
The point of all this is that when we say matter we means anything that scales like matter as the universe expands. So the phrase dark matter means something we can't see that scales as $1/a^3$. We don't know what dark matter is made of, but we do know that whatever it is it has to behave like ordinary matter as the universe expands.
A: Too long for a comment, but on the nomenclature, Fritz Zwicky's observations of other galaxies made it apparent to him that galaxies had to have much more mass than could be seen.   The visible stars added together were about 100 times too light to explain their stellar orbital velocity.   There had to be a lot of unseen mass to hold the galaxies together.  He called this unseen mass "dunkle materie", or dark matter.
Source.
When Zwicky made this claim, the study of other galaxies was in its infancy and and it wouldn't have been unreasonable at the time to assume that Zwicky's "dark matter" was clouds of dust and gas and maybe dark/extinguished stars, some asteroids here and there.  That dark matter was some new kind of undiscovered "particle" wasn't generally agreed to until the 1980s but and by then, it already had its catchy name in place.  
Unexplained mass is a more accurate term.   Dark Matter is a cooler name.   Cool names tend to stick, accurate or not. 
A: When astronomers look at the orbital speeds of stars in galaxies (using the Doppler Shift), those speeds imply there is far more gravity and so far more mass in those galaxies than can be accounted for by ordinary matter.  
The best measurements show that there is about 5 to 6 times as much dark matter as ordinary matter.  The actual amount of dark matter in each galaxy varies.  One galaxy, called Dragonfly 44, is about 99.9% dark matter. 
Astronomers can also measure how much mass is in a region by using 'gravitational lensing'.  Gravity bends light, and so through careful analysis, astronomers can calculate how much gravity and so how much mass is in a particular region.
One of the critical observations using gravitational lensing is of a group of galaxies called the Bullet Cluster.
In the Bullet Cluster, the collision of galaxies has caused the self interacting gas to move differently than the barely interacting stars.  (Stars have such a low number density, that they almost never crash into each other, or even get very close when their galaxies collide.) There is a region of space within this cluster where the gravitational lensing shows there is mass, but there is no ordinary matter to account for it.
Now to the part of your question where you ask about the definition of matter.  Dark Matter has mass, otherwise it would move at the speed of light and not stick around in a galaxy.  Ordinary matter takes up space because of the electromagnetic interactions and Pauli Exclusion. Dark matter is not expected to have electromagnetic interactions with ordinary matter and since dark matter isn't made of ordinary electrons Pauli Exclusion wouldn't apply.
There are some hints that dark matter should be self interacting.  That even though it only interacts gravitationally with ordinary matter, it might have an exclusive interaction with itself.
