In general, you can use the "link" menu item in the files menu to create a link to an object in one file from another file. This creates a non-editable instance of the object in the second file. Alternatively you can use the "append" menu item in the same place to create a copy of the object from one file into another.

However, there are a number of considerations that come into play when trying to combine parts of two files you downloaded. First, while they may be the same scale and orientation, it is possible that they may not. Second, you are using the term "rig", but it's not clear to me that your are referring to the term in the common use in Blender, which is in animation. If so, additional knowledge and skills come into play which are at a somewhat advanced level.

In general, you can use the "link" menu item in the files menu to create a link to an object in one file from another file. This creates a non-editable instance of the object in the second file. Alternatively you can use the "append" menu item in the same place to create a copy of the object from one file into another.

However, there are a number of considerations that come into play when trying to combine parts of the two files you downloaded, among which may be incompatibilities of and orientation of the two objects, and the rig having been designed for specific attachment points that may not also not be compatible between the rig and the mesh.