I've just changed the front brake pads on my 2005 Mazda 6. The new pads are Pagid pads. The pads that I removed have metal plates (shims?) on the back but the new pads don't. The new brakes appear to work fine but now I'm not sure whether I should have taken the metal plates off the first pads and put them on the new set. I'd be very grateful for and advice.

They likely assume the old shims will be reused and properly coated with Molykote M77 or similar product. Shims are mainly used to prevent noise. More important than the shims is that you cleaned and greased the caliper pins with silicone grease. Make sure the new pads are compounded the same as the original pads--don't substitute, for example, semi-metallic for ceramic. Don't forget, also, to change the brake fluid every 2 years. This link may help:

https://www.askmehelpdesk.com/cars-trucks/faq-how-troubleshoot-repair-maintain-hondas-selected-other-vehicles-46563-9.html#post1618425

Wagner Thermoquiet brake pads are the only pads, I am aware of, designed not to use the original shims. Most others assume the shims will be reused.