A connected closed manifold can contain another one as a proper subset: for instance, the $1$-sphere (circle) is contained in the $2$-sphere. Is it possible with manifolds of the same dimension?
Precisely, is there a connected closed $n$-manifold $M$ containing a closed $n$-manifold $N\subsetneq M$?
No. $N$ is open (because of the same dimension), and closed (by assumption), therefore a connected component of $M,$ therefore equal to $M$ by assumption.