Another old wives’ tale (or old husband’s if you prefer). Mayonnaise, because of its relatively low pH (in other words, it is acidic) will actually help prevent spoilage. The pH (level of acidity) of commercial mayo in the US is required to be 4.5 or below, and that is very uninviting to spoilage bacteria. When chicken salad or something similar spoils, it is the other ingredients spoiling, not the mayo. When going on a picnic or setting out a buffet it is important to keep foods cold, but there’s no reason to avoid mayo.
The avoidance of mayo in some situations likely got its start back in the days when mayo was almost always homemade, and homemade mayo typically is a lot less acidic than today’s commercial product.