Two income households are usually households that
depend on two incomes. In other words instead of basing your standard of living on one income you base your standard of living on two. So
both incomes are
required to pay your rent/mortgage/water bill/insurance/car payments/etc.
It is more fragile because 1) it encourages living well beyond your means and 2) the odds of job loss/income reduction is now doubled and the stakes are higher as the second partner can't get a job to pick of the slack in the event of job loss.
It goes back to the old observation that people living paycheck to paycheck on $30,000 tend to also live paycheck to paycheck if their income increases to $60,000.
So if there is a hypothetical one income family making $40,000/year is unable to pay bills instead of reducing spending (who wants to do that?) the other partner starts working. Their lifestyle tends to rise with income and now they are just getting by with $60,000/year and put themselves in a more fragile place. This is how people tend to behave.
Instead they could have reigned in on their expenses a bit and perhaps taken on a second income to save for a rainy day.
The way to do two incomes (if at all possible) is to base your standard of living on just one. Use the other income to invest, save, or fuck around (travel, dinner, etc.). This type of two income house is much less fragile as savings and investments can accumulate, emergencies are easily taken care of, and spending can be reeled in in case of financial heartship. Either way with two incomes savings should be a very high priority to reduce fragility.
So two income households certainly can be less fragile but often aren't in practice.