Roy, it's almost a strictly American thing you are describing. I used to run my Golden Retriever on birds, rabbits, squirrel and it would even tree the odd daytime coon. But when I started getting into coonhounds again if you had a dog that ran anything but coon you had a trashy dog. No possum, no cats, no fox, etc etc. When I switched to terriers I was still kinda stuck in this mindset. I wanted it to run coon and that was it. But after time I came out of my shell. I don't care what my dog runs to an extent. Obviously livestock and deer are off the list, so are skunks. Other than that it's all good. At night I don't care if a $8 coon, a $1 possum or a no account feral cat. During the day coyote, fox, squirrel and rabbits......go for it. After all, what the heck do I care what comes home. Game is game. If i can sell it or eat it I happy.
Over sea's, many dogs including terriers, wire haired pointers, shorthairs, etc are game dogs.