I’ve been watching tutorials in modeling and have come across terms like topology and manifold. These terms are common in mathematics related to studies on calculus, so I was wondering how similar are these concepts to the ones in mathematics, if at all? Thanks.

Those terms are supposed to refer to same mathematical notions.

Topology is relative to the wireframes of meshes in 3D modeling, not to surfaces.

In a polygonal modeler like blender, your 3D model is just a surface.

So, you can create a manifold or a non-manifold 3D model that will mathematically correspond to manifold or non-manifold surface.

By simplification, selecting non-manifold elements of a mesh means selecting mesh elements at a boundary or a self-intersection of this surface.

Actually I still find the different notions confusing, also.

In 3d topology is about mesh connectivity, not about a notion of continuous functions in relation to the surface.

But if you do not use the calculus definition of topology but the more generic set-theoretic definition, the mesh connectivity can be described as a topology on the set of vertices/edges/faces.