"The laws often prevented commoners from imitating the appearance of aristocrats, and could be used to stigmatize disfavoured groups. In Late Medieval cities, sumptuary laws were instituted as a way for the nobility to limit the conspicuous consumption of the prosperous bourgeoisie. Bourgeois subjects appearing to be as wealthy as or wealthier than the ruling nobility could undermine the nobility's presentation of themselves as powerful, legitimate rulers."
Sumptuary laws might have been part of it. But likely it was that natural resources were all highly regulated by the end of the Middle Ages in Western Europe. That was especially true of hunting and fishing rights (even things like taking firewood from a forest could be regulated), so I imagine that keeping animals like pigeons were an offshoot of those kinds of customary rights and laws.
"The laws often prevented commoners from imitating the appearance of aristocrats, and could be used to stigmatize disfavoured groups. In Late Medieval cities, sumptuary laws were instituted as a way for the nobility to limit the conspicuous consumption of the prosperous bourgeoisie. Bourgeois subjects appearing to be as wealthy as or wealthier than the ruling nobility could undermine the nobility's presentation of themselves as powerful, legitimate rulers."