Praise be to Allah.
There is nothing special about the pigeons of Makkah or the pigeons of Madeenah, apart from the fact that they may not be hunted or disturbed so long as they are within the boundaries of the sanctuary, because of the general meaning of the hadeeth: “Allaah had made Makkah sacred, and (fighting therein) was not permitted to anyone before me and will not be permitted to anyone after me. It was only permitted to me for a brief time on one day. Its shrubs are not to be uprooted, its trees are not to be cut down, and its game is not to be disturbed.” (Narrated by al-Bukhaari, 1349). And he SAWS (peace and blessings of Allaah be upon him) said: “Ibraaheem made Makkah a sanctuary, and I have made Madeenah a sanctuary, whatever is between its two lava fields [?]; its trees are not to be cut down and its game is not to be hunted.” (Narrated by Muslim, 1360).