By uniformity, I mean things like postulates, or presumptions. Can we talk about such a thing in the arts? For example, in poetry or literature (as of literary arts) or in visual arts?
Two points :
In any strict sense of 'postulate' the answer is 'no'. Postulates belong to the realm of theory and argument; and this is not what the arts are about. A postulate is an initial assumption; a set of postulates is a system of assumptions from which the propositions of a theory are deducible. I can't think of any counterpart to this in poetry, drama, fiction.
On the other hand, (a) artists often follow a structural plan or ordering of elements in their work. These plans or elements are a kind of postulate. A writer might follow Aristotle's 'rule' (actually a misinterpretation of his 'Poetics') that a play or poem must have a beginning, a middle, and an end each with a distinct, assigned role in the work. Or from a different angle (b) an artist in the former Soviet Union might postulate that all art must observe the standards of 'Socialist Realism' and concern itself with the life of the working class, the peasantry and the intelligentsia.
Taking a cue from Socialist Realism we can track various artistic movements - Classicism, Romanticism, Symbolism, Dadaism, Futurism, Abstractionism and Conceptual Art - which make essential and distinctive assumptions about the role of the artist and the nature of her or his work.
The assumptions in 2. and 3. are not postulates of the sort identified in 1. But there is enough similarity (metaphorical, if you like) for them to count as postulates in a loose but perfectly proper and useful sense.
When trying to discern "unifying principles" to the various art-forms you must rather go to where the art is perceived-conceived → our human mind/brain. The experiences of "art" (high or low) can only be judged and then compared by how it overall effects our mind-brain. Thus, "beauty is in the mind of the beholder." Unfortunately most stop here with it's only your opinion but really should be the beginning for yourself to find how your mind-brain "constructs" itself in response to art. This is nothing short of some deep self-psychoanalysis. Even more problematic is trying to share your insights with someone else which requires them to duplicate your process. That so few can so engage explains the pitiable state of aesthetic analysis and criticism. Thus said, a worthwhile pursuit.