There are some levels of confusion in this in question, but basically we can test
- what wavelengths animals perceive (simple behavioral tests, e.g. training & testing them to distinguish a certain wavelength)
- inspect their cones to determine what wavelengths they could perceive. The latter doesn't necessarily imply that if an animal has say 4 different cones, it has tetrachromatic vision, because there's some degeneracy in some species, which unlike in humans (which have 3 types of cones and trichromatic vision) don't really use all their cones.
To give you a nearly random example of such a paper (this species doesn't have degeneracy), a (freely available) paper on chicks' vision:
The colour vision of domestic chicks (Gallus gallus) was
investigated by training them to small food containers
decorated with tilings of grey and coloured rectangles.
Chicks learn to recognise the colour quickly and
accurately. Chicks have four types of single-cone
photoreceptor sensitive to ultraviolet, short-, medium- or
long-wavelength light. To establish how these receptors
are used for colour vision, stimuli were designed to be
distinguished only by specific combinations of receptors.
We infer (1) that all four single cones are used, and (2) that
their outputs are encoded by at least three opponency
mechanisms: one comparing the outputs of ultraviolet- and
short-wavelength-sensitive receptors, one comparing the
outputs of medium- and long-wavelength receptors and a
third comparing of the outputs of short- and long- and/or
medium-wavelength receptors. Thus, the chicks have
tetrachromatic colour vision. These experiments do not
exclude a role for the fifth cone type, double cones, but
other evidence suggests that these cones serve luminance-based tasks, such as motion detection, and not colour
If you want to get philosophical qualia discussions, akin to whether all humans describe the same color with same language, that's a whole different kettle of fish. I don't know of any studies like that in the absence of language. I'm guessing you might be familiar with the dress.