On the Wekipedia website Configuration space , there are two configuration spaces defined. One is Configuration spaces in physics, the other is Configuration spaces in mathematics.
Question. Do these two configuration spaces have any connections or essentially the same meaning? Or they just happen to share the same name?
No, they do not essentially have the same meaning. As stated in the Wikipedia lemma, the 'mathematical' notion of configuration space is closely related to the state space in physics. On the other hand, they don't 'just happen to share the same name', because the notions behind the constructions are similar in some way.
To be very brief, the physical notion of 'configuration space' is the most restrictive, since it has a clear physical meaning. That is, they represent the position of a particle, or the position and orientation of a rigid body, both in physical space. The 'mathematical' configuration space, on the other hand, can be (and often is) a much more abstract concept. For example, the different states a quantum-mechanical system can be in can be considered to live in a certain 'state space', which is often a type of Hilbert space. The interpretation of an element of such a state space is often not as direct as a point in the 'physical' phase space. For the quantum-mechanical example, a point in the state space is a wave function, the physical interpretation of which is a whole research field in itself.