A manifold is an abstraction of Euclidean space, so it has less properties, and allows for more.
That is, until we add extra properties to a manifold that are not possible in Euclidean space.
A manifold is only locally homeomorphic (has a bi continuous function) to Euclidean space $\mathbb R^n$. Homeomorphic means that there is a continuous 1-1 mapping that has an inverse that is also continuous.
It means that a manifold does have angles and distances, but these can change from point to point in a continuous fashion.
As such it allows:
Curved spaces since the global parallel postulate of Euclidean geometry is not (necessarily) true.
Non-orientable shapes like the Klein bottle.
Disconnected components that may even have different dimensions.