I have simulated a DSGE for the same time and Dynare shows me a result I cannot fully understand.
In the correlation matrix, it shows me there is a perfect correlation between two variables, but however, it still works. Shouldn’t Dynare tell me I have a singular matrix?

Why should it? Singularity of the covariance matrix of endogenous variables is not a problem.
Problems only arise if matrices are singular that need to be inverted.