A vertex operator algebra describes the algebra of local operators in the chiral part of a 2d CFT. Typically one sees a VOA depending on a complex coordinate $z$. To describe a full 2d CFT, you would need to also include an "anti-chiral" VOA depending on a conjugate coordinate $\bar{z}$. So by considering only a single vertex algebra depending on one complex variable, you are only considering a "chiral half" of the CFT.
In physics more generally, people will often refer to "chiral algebras" of local operators in other types of theories of different dimensions as well, so in that field the terminology is quite broad, but relates to the chirality of a theory in a traditional sense.
Mathematicians have since extended and generalized a number of things relating to vertex algebras. For example Beilinson-Drinfeld chiral algebras generalise vertex algebras, the Chiral de Rham complex and chiral differential operators are VOA versions of differential forms and differential operators etc... In the past, constructions such as these were the only examples of mathematically well-defined and well-studied chiral algebras (in the physics sense of algebras of local operators). Hence in the mathematical community, I suspect it became practice to name constructions relating to vertex algebras and BD chiral algebras "chiral" since this was the only real example of a chiral algebra of local operators they were looking at in those communities.
In Arakawa's work, "chiralization" essentially means going from an associative algebra type object to a VOA type object. More generally, I believe chiralization/adding the word "chiral" to something in these communities will describe working with something like an affine or loopy version of a previously known construction.
For example, taking the Weyl algebra of differential operators, one can consider its VOA counter-part which is the $\beta\gamma$ VOA, also known as "chiral differential operators". The modes of this VOA satisfy similar relations to the Weyl algebra and the Weyl algebra can be recovered from it in various constructions.
I guess the point to emphasise though is that the word "chiral" itself is often not-literal in the mathematics community, but is rather describing something like "VOA-ization" (which in BD language is chiralization).