$\newcommand\tk{\tilde k}\newcommand\ip[2]{\langle #1,#2\rangle}$Let $k$ be a reproducing kernel of a [reproducing kernel Hilbert space (RKHS)][1] $H:=\mathcal H$ of real-valued functions on a set $X$. Then 
$$\ip f{k_x}=f(x)\tag{1}$$ and 
$$k(x,y)=\ip{k_x}{k_y}=k_x(y)\tag{2}$$ 
for all $f\in H$ and all $x$ and $y$ in $X$, where $\ip\cdot\cdot$ is the inner product on $H$. 

Let now $\tk$ be another reproducing kernel of $H$. Then, by (1) and (2), for all $x\in X$
$$\begin{aligned}
\|k_x-\tk_x\|^2& =\ip{k_x}{k_x}+\ip{\tk_x}{\tk_x}-\ip{k_x}{\tk_x}-\ip{\tk_x}{k_x} \\ 
& =k(x,x)+\tk(x,x)-k_x(x)-\tk_x(x) \\
& =k(x,x)+\tk(x,x)-k(x,x)-\tk(x,x)=0,
\end{aligned}$$
whence $k_x=\tk_x$ for all $x$, that is, $k=\tk$. 

(You mismatched colors.) 
  
  [1]: https://en.wikipedia.org/wiki/Reproducing_kernel_Hilbert_space#Definition