Can household dish soaps, stored diluted, breed dangerous bacteria

The USDA has this to say on it:

Storing Leftovers
One of the most common causes of foodborne illness is improper cooling of cooked foods. Because bacteria are everywhere, even after food is cooked to a safe internal temperature, they can be reintroduced to the food and then reproduce. For this reason leftovers must be put in shallow containers for quick cooling and refrigerated within 2 hours.

You'll find similar statements from government agencies around the world. The safe limit for raw or cooked food is 2 hours in the danger zone (40-140° F or 4.4-60° C).

If you're a restaurant owner or cook, you must follow this rule, hold hot foods above 60° C and quick-cool other foods before refrigerating. If you are not working in a professional capacity then you are not legally required to follow it, but if you are serving guests then it would be irresponsible (and possibly actionable, if someone gets sick) to do otherwise.

If you're an individual serving only yourself, then take whatever liberties and break whatever rules you want; it's your food, and your body. But there's no table or chart anyone can give you; there's no single specific point at which a food transitions from "not entirely safe" to "probably will kill you" because it depends entirely on the food, the environment, your immune system, and a plethora of other variables. The rule is 2 hours, period; any longer and there is some non-trivial risk to your health.

Some hints, tips, and warnings:

• The 2-hour rule is a conservative estimate with a safety margin. Don't ask what that margin is. It's like asking what the "real" speed limit is on a posted road; you might know from experience, but it could change depending on circumstances and exceeding it by any amount means you take your chances and accept the risks.

• Don't put large, hot items (such as an entire pot of soup or chili) directly into the fridge. The residual heat will warm up and potentially spoil other items in the refrigerator.

• To quickly cool large cooked items, use an ice-water bath and/or divide them into small containers. (Note: Don't use an ice-water bath for cast iron.)

• Don't assume that re-cooking an improperly-stored item will make it safe. Most bacteria produce protein toxins, which are actually the primary agents responsible for food poisoning, and several of these toxins are heat-resistant. Cooking will not kill or inactivate these toxins and eating the re-cooked food will still make you sick.

• Don't assume that cooking "kills everything" and that a cooked food or cooking surface is absolutely sterile. Cooking kills enough to make the food safe to eat, but some organisms - such as bacterial spores from bacillus and clostridium - can survive the cooking process and immediately start producing more bacteria. Sous-vide bags, crock pots, etc. are not safe environments for cooked food in the temperature danger zone.

This is not an argument you can win. Say your prepared food item is a raw carrot. Clearly carrots are safe at room temperature for weeks and months. They might get rubbery and unpleasant, but they aren't going to make you sick. You can probably even prove this. Now consider a bowl of chicken stock. Rich in both nutrients and water, bacteria are going to love growing in it and even an hour or two at room temperature might be dicy. And between those two things you have items like baba ganoush which, yeah, is probably ok for a few hours. But that's not the point.

Inspectors and rules can't do "spectrum" and they can't do "read the ingredient list" or "read the ingredient list and cooking instructions" (since raw beaten eggs would poison you way faster than a hardboiled egg would.) They need a simple rule that inspectors and vendors alike can understand and enforce. The actual facts of how that particular dish spoils do not and cannot come into it. Customers, vendors, and inspectors need to see at a glance "yes, this place is following the rules" not "well, that might be ok but I'll need to see whether or not they have icing sugar in this version".

It's not about what would be safe at home. It's not about the science of bacteria growth. It's about overall consistency and overall safety. A beautifully-optimized set of rules with 10 or 20 or 50 categories might feel fairer, but would confuse entry-level employees, require tons of signs, labels, written procedures etc, and a vendor would never be completely confident that an inspector wouldn't say "this isnt A-1B dip, it's clearly A-2B dip and I'm writing you up!"