Take three tetrahedron and glue them along one of their faces. With the obvious simplicial complex structure this is what you are interested in since every edge is incident to either 2 or 4 faces. This is not the result of taking a disjoint union of manifolds with a simplicial structure and identifying vertices with vertices and edges with edges because we can split in to two cases:
It is a quotient of multiple manifolds:
If this were a quotient of k connected manifolds, then we would be able to partition our 2-simplices into k disjoint sets each of which has the property that their sum is a cycle. These sets are just the faces of each connected manifold.
This clearly can't be the case here since any cycle with a face of one of the tetrahedron in it must have the entire tetrahedron.
It is the quotient of a single manifold:
This cannot be the case because if we look at the shared 2-simplex and pick an edge, we could find which face was adjacent to the shared simplex, along this edge, prior to gluing. Say it is from the first tetrahedron. We can see our manifold must contain a copy of this tetrahedron since away from the shared simplex, every point has a Euclidean neighborhood. However, this tetrahedron can't possibly be the initial manifold since we are missing many faces. But we run into a contradiction since no connected manifold could have a copy of $S^2$ in it as a proper subspace.
We conclude that there is no manifold with a simplicial complex structure such that it has a quotient retaining its simplicial complex structure which is the simplicial complex we constructed.