How to connect 2 lamps to one switch wirelessly
You must do your shopping for smart home devices.  Specific product recommendations are off-topic on this Stack Exchange (although I did give a specific example in a comment).

What you’re looking for tho is a wall switch that acts as a controller for one lamp, and a plug in module for lamp 2.   You don’t need to sense voltage at all, you just need a smart switch that will turn on another device as well as itself.  In some cases you may or may not need a particular systems hub.  I would also watch out for systems where internet connection is required for the setup to operate correctly.