I don't think it matters much if they match. Depending on how much of each you do though, you could probably accomplish the same with one rifle. Something like a .243 or .260 Remington would work. Even if you want to stick with .308, you can use the 110-gr or 130-gr loads for vermin. In deer hunting, you're only likely to make one or two shots on a particular hunt, and for coyote hunting, I'd guess an extremely productive day might involve 10 shots? A .308 doesn't kick so bad that you couldn't shoot it 10 times or even have a nice day at the range. It isn't like you are setting up on a groundhog colony and planning to shoot 100 times in a day in which case the .223 makes more sense.
Just thinking out loud, but if you stick with one gun, you can get a nicer gun, nicer optics, and plenty of ammo for what it would costs you to get two similar rifles setup. The $1000 or so it costs to setup a nice rifle would buy a good bit of .308 paper punching ammo.
A slightly different option might be to get an AR-15 with a .223 upper and a 6.5 Grendel upper. That might not save you much over a pair of bolt guns though, but it would be the same shooting experience more or less.