Consider a circular loop falling in a uniform magnetic field as shown. By Faradays law, as it falls there is no change in flux, so no induced emf. But, if we apply Motional EMF equations the the 2 semicircles, emf becomes $$B(2r)v$$. Why is this happening?

The induced motional emf across the ring will be $$B2rv$$ as you have stated with the "top" part of the ring and the bottom part of the ring contributing the same magnitude.
The rails together with the termination $$PQ$$ define another loop with only the loop moving and the rails and termination stationary and the magnetic flux through that loop is changing.