I think the answer to your question is no, and it follows from the following paper of Koppelberg and Shelah Subalgebras of Cohen algebras need not be Cohen.

In this paper, it is shown, for each $\kappa \geq \aleph_2,$ there exists a non-Cohen complete subalgebra of $Add(\omega, \kappa)$.

This subalgebra, is c.c.c and strongly proper, as a projection of Cohen forcing, but is not isomorphic to Cohen forcing.

The answer to your question is yes, and it follows from the following paper of Koppelberg and Shelah Subalgebras of Cohen algebras need not be Cohen.

In this paper, it is shown, for each $\kappa \geq \aleph_2,$ there exists a non-Cohen complete subalgebra of $Add(\omega, \kappa)$.

This subalgebra, is c.c.c and strongly proper, as a projection of Cohen forcing, but is not isomorphic to Cohen forcing.