Isomorphisms are absolutely essential to category theory, and in particular the idea that isomorphic objects are "the same" is perhaps the single most important concept in all of category theory. As you observe, you can define isomorphisms without requiring the existence of identities. However, this definition is still a bit problematic in a few ways. First, if "isomorphic" means "the same", you would expect it to be an equivalence relation, which it is not if identities don't exist. Second and more importantly, functors do not have to take isomorphisms to isomorphisms if you do not require that they preserve identities. This is a very serious problem for the intuition that functors should be some kind of "natural" operation: intuitively, an operation that can send isomorphic objects to non-isomorphic objects seems like it would have to be highly unnatural. I suspect this had at least something to do with Eilenberg and Mac Lane's decision to require that categories have identities (and that functors preserve them).
Here is another very basic thing that can go wrong in the absence of identities: you can have non-isomorphic terminal objects. Let $\mathcal{C}$ be a category with three objects $A$, $B$, and $C$ and exactly one map between any two objects, and let $\mathcal{D}$ be the non-unital category obtained by adjoining to $\mathcal{C}$ a new map $f:A\to C$ whose composition with any map of $\mathcal{C}$ is the unique map in $\mathcal{C}$ with the same domain and codomain. Then $A$ and $B$ are both terminal in $\mathcal{D}$, but they are not isomorphic (in fact, $A$ has no identity map).
Since any sort of universal property can be described as a terminal object in an appropriate category, this means that in non-unital categories, you cannot expect any universal property to define objects uniquely up to isomorphism. One way to fix this is to modify the definition of "terminal" to require that a terminal object have an identity map. Another way is to just require that all objects have identity maps, which holds in pretty much every example of interest.
Finally, on a more philosophical note, I would say that the existence of identity elements is actually a corollary of the "correct" definition of associativity. I would say that an "associative operation" is an operation which given any finite ordered set of elements $(a_1,\dots,a_n)$ gives you a "product" $a_1\dots a_n$ such that you can always drop parentheses (so for instance, $(ab)c=abc$, where the left-hand side denotes the binary product of (the binary product of $a$ and $b$) and $c$, and the right-hand side denotes the ternary product of $a$, $b$, and $c$). The identity element is then just the product of the empty ordered set. Considering "associative" operations which don't have identities is thus analogous to considering finite sets but not allowing the empty set. Of course, this is sometimes useful to do, but unless you have a good reason to, it is probably not the natural thing to do.
Best Answer
There's one fundamental example you should add to your list: functions from sets to sets.
The number of functions from a finite set $A$ to another finite set $B$ is $|B|^{|A|}$, where $|A|$ and $|B|$ are the number of elements in the sets. This is the origin of the notation $B^A$ for the set of functions from $A$ to $B$. Most of your examples expand the analogy to another situation.
$C^X$ for $C$ category and $X$ set. Here $C^X$ is the set of functors from $X$ to $C$. Since $X$ is a set it can be treated as a category with its only morphisms the identities from each element to itself. Functors from such a discrete category reduce to functions from $X$ to the collection of objects of $C$.
$c^{b\times b'} \cong (c^b)^{b'}$ for $c$, $b$ and $b'$ objects. Here, we're using the notation as an internalization of the set of morphisms from $b$ to $c$. Rather than a set, we can sometimes have objects in the same category that act like the hom sets.
$G^X$ for $G$ group and $X$ category. This is similar to the first example. Here, $G$ is made into a category by giving it a single object and making the morphisms from that object to itself the elements of $G$. Composition is just the group operation.
$F^T$ for $F$ and $T$ both functors. Here we're using the usual category structure on the collection of functors between two fixed categories. The objects are functors and the morphisms are natural transformations. So $F^T$ is the set of natural transformations from $F$ to $T$.
One other example doesn't quite fit into the same vein, but is still closely analogous.
The last two examples, I'd need more context or I'm not familiar with the origin of the notation.
$X^T$ for $X$ category and $T$ functor. I believe this is the category of $T$-algebras. A $T$-algebra $A$ has (as part or all of the data, depending on whether we're talking about algebras of a monad or algebras of a plain functor), a morphism $T(A) \to A$. So there's some notion that there are "morphisms from $T$ to (particular objects of) $X$", but I'm not sure that that's why this notation is used here.
$\eta ^T$ for $\eta$ natural transformation and $T$ functor. I'm not familiar with this usage.
It may be worth noting that some of the notation in Categories for the Working Mathematician is outdated so you shouldn't use it as your sole source. Read other books and papers to see what notation other people use for these concepts.