I see that even if the home owner insurance is not with the same carrier as Auto Insurance but still auto insurance company gives the home owner discount. And if it is with the same company then it is called "multiple policy" discount.
Also, One of the person has been renting and has the car garage, but that person did get the discount for garaged car but it was less than the home owner discount.
How does the probability of accident or other loss decrease with home ownership?
Well, when it's with the same company, it's all about policy retention - if you have both your auto and homeowners with the same company/agent, you're more likely to renew them both, with the same company. If you only have one, the other agent is going to try to take that other policy from you.
Regarding a homeowners discount for homeowners, that really is a loss ratio thing. If you can be responsible enough to manage to own a home, then that responsibility generally translates into driving slightly more responsibly, or at least having fewer claims dollars paid out.
You CAN get the credit just as equally, though, with a RENTERS policy as with a homeowners policy. Usually, that credit is a percentage of the premium.
Insurance companies don't worry about how, or why, there are fewer claims with home ownership - just like they don't worry about how or why with credit scores, marital status, gender, etc. The how or why isn't relevant. All that's relevant, is that there is a direct corrolation between those items, and claims. The reasons behind it are irrelevant.
"How does the probability of accident or other loss decrease with home ownership?"
This is the wrong question. Auto insurance is priced on correlations rather than causations. There appears to be a correlation between lower loss costs on auto insurance for people who have homeowners insurance when compared with the entire cohort of people with auto insurance.