Consider two random variables $X$ and $Y$ uniformly distributed on $[-1,1]$, and $P(X = Y) = 0.5$. Their correlation coefficient is $\rho_{XY} = 0.5$.
What is the meaning of $P(X = Y) = 0.5$ here? Is this have any relation with the correlation coefficient?
$\mathsf P(X=Y)=0.5$ means that there is the probability of $0.5$ for $X$ and $Y$ to realise the same value.
Now their covariance is $\mathsf {Cov}(X,Y) ~{= \mathsf E(XY)-\mathsf E(X)\mathsf E(Y) \\ =\mathsf E(XY) \\ = \tfrac 12 \mathsf E(X^2\mid X=Y)+\tfrac 12\mathsf E(XY\mid X\neq Y)}$
We cannot really say much more unless we know where $X$ may equal $Y$ and how they may be related when they are unequal.
However, it is not necessarily so that their correlation is $0.5$, unless there is some added context which you have not mentioned.
For instance, suppose $Y=\begin{cases} X &:& W=1\\ -X &:& W=0\end{cases}$ where $W\sim\mathcal {Ber}(1/2)$ and $X\sim\mathcal{U}[-1;1]$, independently.
Then $Y\sim\mathcal{U}[-1;1]$ and $\mathsf P(X=Y)=0.5$ are desired, but the variables are in fact uncorrelated. $$\mathsf{Cov}(X,Y)~=~ \tfrac 12\mathsf E(X^2\mid W=1)+\tfrac 12\mathsf E(-X^2\mid W=0) ~=~ 0$$
However, if we let $X,Z\sim\mathcal[-1;1]$ and $W\sim\mathcal{Ber}(1/2)$, mutually independently, and define $Y=\begin{cases} X &:& W=1\\ Z&:& W=0\end{cases}$, then indeed $\rho_{X,Y}=0.5$. (Since $\mathsf{Var}(X)~=~\mathsf{Var}(Y)~=~\tfrac 23$.)
$$\mathsf{Cov}(X,Y)~=~ \tfrac 12\mathsf E(X^2\mid W=1)+\tfrac 12\mathsf E(XZ\mid W=0) ~=~ \tfrac 13\\ \rho_{X,Y} ~=~\dfrac{1/3}{\sqrt{(2/3)^2}}~=~ \tfrac 12$$