Yes it is possible.
Assuming you have done [no] custom weight painting. You can simply achieve the initial starting base pose state, delete the old armature modifier, and parent the new armature with automatic weights.
Deleting the armature modifier of the mesh wont delete the animations of the armature. Armature modifier is different than the armature itself. The armature is referenced in the armature modifier. You can even parent the armature to 2 or more meshes.
To achieve a base position you can clear all the rotations and positions temporarily. There are menu options such as [clear rotation] for bones of an armature. Consider having frame 0 for that case. You can render frames 1 to N which does not include frame 0.
If you are planning to replace one mesh entirely with the new mesh simply achieve your base position and parent the armature to the new mesh. Test. Then delete your old mesh. You may want to save multiple copies of your Blender file. Blender allows you do make any mesh invisible to view or render which may be convenient in your case.
If you have done custom weight painting. Just go into weight paint mode and paint the weight for the new bones and test.
You picture appears to introduce a new question. If you have two meshes you can join them with a menu selection [Join Meshes]. Makes sure you select the dominant mesh last.
If the meshes are two different objects you do not even need to join them. The single armature can be the target of more than one armature modifier on two different meshes as stated above.