Just to correct two mistakes at the beginning:
Not. There are very few people interested in it, and organizations like that are fringe.
No. This simply is not the case in countries without such laws (including several European ones) and did not happen in Britain before Dangerous Animal Act was passed.
Currently, there are no problems with escaping wild animals. Stories that in Texas there are hundreds of tigers kept in backyards and in New York there is a population of escaped alligators in sewers are urban legends. So there is no problem to solve.
Dangerous wild animals on the loose were also not a problem in Britain before this law was passed. And, as others pointed, the current law is already bloated. It covers venomous reptiles which venom is not dangerous to humans, carnivores no bigger than a domestic cat and wild bovids smaller and more docile than a domestic cow or goat.
Do you know that, according to British law, a dikdik is dangeorous wild animal? And gazelle, too? Just lumped with all
Bovidae.*
Owners commonly point that money required to comply to bureaucracy goes directly off money for care of animals themselves.
Fringe organizations which propose such laws often have habit of creating pointless controversies to promote themselves, and often have a hidden animal rights agenda which they try to smuggle by other means.
There is little possible benefit of such regulations, other than unnecessary feed bureaucracy and give activists control other people. If anything, dangerous wild animal law should be reviewed and become less bureaucratic.
*
Dangerous Wild Animals Act 1976 - Wikipedia