That sounds bizarre to me. A nursery, or any other organisation may not like complaints but it's in their own interests to hear them. Otherwise the complainer might complain somewhere else, like writing a letter to a newspaper, or posting in a forum, or to a lawyer who might then advise the complainant to file a case. I've never heard of anything that legally prevents someone from making a complaint to a company (and any sensible company would welcome complaints since it is free feedback that saves them paying a fortune to a survey consultant for the same information).
Of course, there are laws to protect organisations and individuals against slander or libel, so maybe it's possible Emerald City Nursery is feeling libeled or slandered and want to have a crack at the complainer for that, rather than view the complaint as a useful piece of information that Emerald could use to improve their services. The other explanation I can think of is that ECN feel that what is being complained about is deserving of legal investigation to protect themselves, for example if a parent complained that something had been stolen, or their child was seriously injured, or ECN had been negligent in some way, or similar, however it doesn't sound like that's what you're talking about.
Emerald City Nursery (ECN) has been added to the
Dubai nursery list (thanks for pointing that out C)

.