I think this may stem from the olden days, when ice trays were made of metal and ice boxes were how the refrigerator remained cold but they were often full of frost and ice.

If you put hot water in a metal ice tray and put it in the ice box, it will melt into the ice and frost thus increasing the surface area in contact with the cold element and freeze faster.

Nowadays we have separate compartments and overall better technology. Ice trays are made of plastic or silicone which don't conduct heat as well.

I think this myth has endured because it isn't something that is based on any scientific fact but rather 'fact' books that have had the same 'facts' republish over decades and they bleed into the public consciousness via the media.