Is the notion of a 2-category introduced to fix/forget the size issues in the definition of (an ordinary) category? A category $\mathcal{C}$ consists of pair of classes $(\mathcal{C}_0, \mathcal{C}_1)$, along with maps $$\mathcal{C}_1\times_{\mathcal{C}_0}\mathcal{C}_1\rightarrow
\mathcal{C}_1\rightrightarrows \mathcal{C}_0\rightarrow\mathcal{C}_1.$$
A category is called small, locally small, large, depending on the size of the class $\mathcal{C}_1$.

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*If $\mathcal{C}_1$ is a set (which would imply $\mathcal{C}_0$ is a set), then, $\mathcal{C}$ is called a small category.


*If $\mathcal{C}_1$ is a proper class, but $\mathcal{C}(a,b)$ is a set for each $a,b\in \mathcal{C}_0$, then, $\mathcal{C}$ is called a locally small category.


*Anything that does not fall in above two cases is called a large category.
A $2$-category is described to be something, that comes with a class $\mathcal{C}_0$, a category $\mathcal{C}(a,b)$ for each $a,b\in \mathcal{C}_0$, some more data satisfying some conditions.
If I start with a category that is not locally small, I know that, $\mathcal{C}(a,b)$ is not a set. Is it then a "better idea" to consider it as a category, along with some compatibility conditions, instead of "feeling bad" that it is not a set?
Is this how the notion of a $2$-category came into picture?
Is the size issue, in any way, related to the necessity of introducing the notion of a $2$-category?
Does the same justification hold in introducing the notion of an $n$-category?
If none of the above has a fairly positive answer, is there a way to introduce the notion of a $2$-category as an outcome of trying to fix the size issue in ordinary category theory? Is there a way to introduce the notion of a $n+1$-category as an outcome of trying to fix the size issue in an $n$-category?
 A: Historically, the introduction of higher categories has nothing to do with set-theoretic size issues to my knowledge. Promoting the hom-sets of a non-locally small category into (presumably discrete) categories doesn't add any additional information, since there wouldn't be any interesting new data in these categories; as objects they would have the arrows of the hom sets, and their only 'arrows on arrows' would be identity arrows. We could then speak about functors into/out of them, but I don't see what that gets us off the top of my head. See below for an account of how we can naturally arrive at the definitions of higher categories, but that seems secondary to your question.
The 'size-issue' that you bring up is something that occurs when we try to consider a 'thing of all things', without delineating between the first usage of the word 'thing' and the second usage of the word 'things'. The classic example is  Russell's paradox which arises when we try to consider the set of all sets, and this leads mathematicians to believe that it is a set-theoretic issue, but in type theory we have the analogous Girard's paradox when we try to consider the type of all types.
This can be rectified in set theory by considering a hierarchy of collections like sets/classes/super-classes etc., or by using a method like Grothendieck universes and considering a hierarchy of universes
$$V_0\subseteq V_1\subseteq V_2\subseteq\dots$$
Analogously, in type theory we can rectify the paradox by considering a hierarchy of types
$${\sf Type}_0:{\sf Type}_1,\ {\sf Type}_1:{\sf Type}_2,\dots$$
In any case the paradox seems to arise when we try to treat 'all things of a certain nature' as 'a thing of that same nature', in a certain sense that is independent of sets or types or whatever formalism we choose. I do not think there is any need for 'size envy' towards set theorists :). There are foundations like NF that allow for a set of all sets by baking the required stratification into the axiom system (in this case the comprehension axiom is stratified), but they have other strange properties -- in NF for example, the category of sets is not Cartesian closed.
In order to answer your last question, what do you mean by 'fixing' the issue? There are advantages and disadvantages to all the different approaches to rectifying this paradox, and which one a person uses depends on the needs at hand. Do you want a formalism that lets you talk about 'all the things of a smaller nature', like with classes/universes/type hierarchies? Do you want a formalism that makes it so the 'things' are naturally stratified, like NF? An answer to this question would help us maybe think of some natural formalism using higher categories, but I think it's doubtful anything 'canonical' will arise.

Although the following account is probably not historically accurate, it is a canonical path up the abstraction ladder from categories to higher categories.
Beginning with a notion of set, it's natural to define a notion of function between sets -- when combining these notions into one structure, we naturally arrive at the definition of a category.
Now that we have categories we can naturally define functors between them, but the increased depth in the definition of a functor allows us to canonically define a notion of natural transformation between functors. If we want to collect the notions of a category and functor and natural transformation into a single structure, we naturally arrive at the definition of a $2$-category.
Now that we have $2$-categories we can naturally define $2$-functors and $2$-natural transformations between $2$-functors, but the increased depth in the definition of a $2$-natural transformation allows us to canonically define a notion of modification between $2$-natural transformations. If we want to collect together notions of $2$-categories, $2$-functors, $2$-natural transformations and modifications together into one structure, we naturally arrive at the definition of a $3$-category.
Rinse and repeat to move higher up the (strict) higher-category ladder, and weaken equality (down to iso/equivalence/adjunction/$n$-cell etc.) as necessary for pseudo/quasi/lax etc. versions.
