If the variance of two variables are equal, then the covariance is the variance? If I have two random variables $X$ and $Y$ and their variance is the same, i.e.,  $Var(X)=Var(Y)$.
Does this mean $Var(X)=Var(Y)=Cov(X,Y)$ ?

In short, no.

Consider the counterexample that if two normal variables both have unit variance, their covariance could be zero.

Alternatively, for geometric motivation, consider that two vectors can have the same length while pointing in perpendicular/orthogonal directions.
