My trial is : I guess that the identity is the constant loop, which I can denote it by $ C(t) = p$ for all $0 \leq t \leq 1$.
But I know that the path C is an identity iff $C.a = a$ and $b.C = b$ whenever $ C.a$ and $ b.C$ are defined. So my question is why we are taking in the first equality of the definition a path called a while in the second equality we are taking a different path called b?
Also, how to verify the existence of an identity element for the fundamental group by the definition I mentioned above ?
The fundamental group $\pi(X,p)$ does not consist of $p$-based loops, but of equivalence classes of such loops with respect to the relation $\simeq$ defined by fixed-endpoint family of paths. This is a sort of homotopy between paths (but in contrast to ordinary homotopy it also relates paths with different stopping times).
The identity path $e_p$ at $p$ is a $p$-based loop and we know that for any $p$-based loop $a$ we have $e_p \cdot a = a$ and $a \cdot e_p = a$. This transfers to equivalence classes: We have $[e_p] \cdot [a] = [e_p \cdot a] = [a]$ and $[a] \cdot [e_p] = [a \cdot e_p] = [a]$. However, instead of $e_p$ you may also take any constant path $c$ at $p$ ($c : [0, \lVert c \rVert] \to X, c(t) = p$). This is true because $[e_p] = [c]$.
The benefit of working with equivalence classes is that $\pi(X,p)$ becomes a group. If we work on the loop level, then multiplication of loops leads to addition of stopping times and thus never produces an identity path unless both factors are identity paths.
Remark:
It should be mentioned that the approach by Crowell and Fox differs from that in most other books. Usually a path in a space $X$ is defined as a continuous map $a :[0,1] \to X$ defined on the unit interval $[0,1]$. In the sense of Crowell and Fox this is path with stopping time $1$. For the sake of distinction let us denote such a path as a normal path (it is just an ad hoc notation, you will not find it in the literature). This standard approach has disandvantages and advantages. The main disadavantage is that the product $a \cdot b$ of normal paths obtained by "putting $b$ after $a$" as Crowell and Fox do is a map defined on $[0,2]$ which is no longer a normal path. Thus you define $a * b$ (which is again an ad hoc notation to distguish it from $a \cdot b$) by $$a * b = \begin{cases} a(2t) & t \le 1/2 \\ b(2t-1) & t \ge 1/2 \end{cases} $$ which is a normal path. This operation is not associative and does not have identities. Thus Crowell and Fox's approach behaves better in that respect.
The advantage is that the equivalence of normed paths is defined simpler: For normed paths $a, b$ such that $a(0) = b(0) = p_0$ and $a(1) = b(1) = p_1$ we write $a \simeq_* b$ if there exists a homotopy $H : [0,1] \times [0,1] \to X$ such that $H(t,0) = a(t), H(t,1) = b(t)$ for all $t$ and $H(0,s) = p_0, H(1,s) = p_1$ for all $s$.
It is an easy exercise to show that for each path $a$ there exists a normed path $a_*$ (with the same initial point and the same terminal point as $a$) such that $a \simeq a_*$ and that for normed paths $a_*, b_*$ one has $a_* \simeq_* b_*$ iff $a_* \simeq b_*$. Thus on the level of equivalence classes both approaches produce the same result: We have a $1-1$-correspondence between equivalence classes of paths $[a]_\simeq$ and equivalence classes of normed paths $[a_*]_{\simeq_*}$.