Many animals which react badly to seeing their reflection are doing so because they do not recognise the reflection is them. Your cat sees itself in the mirror and thinks: Eek! There's a stranger in MY living room!
Smart animals (people, chimps, elephants, parrots) figure out it is themselves they are seeing. Dumb animals will run away or attack it, but eventually work out that the reflection doesn't actually DO anything and ignore it. Really Dumb Animals (Siamese fighting fish?) will just keep attacking it and attacking it...
If the Earth had been covered in mirrors, then natural selection would favour creatures which were Smart or Dumb. Telling the difference between a mirror and a real member of your species will be a very important skill. They Really Dumb ones will go through their (short) lives wasting energy reacting to reflections. They won't have as much resources to dedicate to other stuff like breeding and defence, and will be outcompeted.
Every spring there will be a rash of newborn animals encountering mirrors for the first time, and figuring out what is real and what is a reflection. Much like baby birds have to figure out that glass is transparent but solid.
Meanwhile, predator-prey interactions become... complicated. Prey are going to get adept at using mirrors to spot predators coming. This includes working out that if the tiger is reflected in that mirror there, it means it is really approaching from the west, so you should run east, right past the reflection.
Predators may have to pick routes without mirrors to sneak up on things. Or may exploit the mirrors to startle prey into the waiting teeth of another pack member.
Animal displays will be magnified by mirrors - one peacock will look like a room full of them. So visual displays may become less important. Scent and sound may replace bright feathers, stotting and strutting.