First of all, these are all gross income numbers and include no savings at all. If your estimated gross living wage is 70k, but you're making 200k, that means an extra 130k. About 55k of that goes towards taxes (see tax rates below), leaving 75k. That needs to cover all your savings (if you make 200k, you better be saving about 20-30k/year for retirement, plus you might want to have college savings and such if you have kids) plus discretionary spending. If you figure $10k/year for college savings for 2 kids (probably low!), that leaves 35-45k, or 3-4k/month in discretionary spending, which is still quite a bit, though nowhere close to the naive 10k/month you mention.
Looking at the sources of these various numbers, the $91k number in the busnessinsider article comes from http://www.epi.org/resources/budget/ which breaks down the annual costs as follows, annually, for the SF metro area:
* Housing: $23k (seems fairly low for a family of 4 in the Bay Area, but I guess it depends on how far out you live)
* Food: $9384 (quite reasonable)
* Child Care: $10815 (seems unrealistically low for 2 kids if both parents work; hard to tell if only one works, since I would expect that to be near-0 for a bare minimum).
* Transportation: $7300
* Health care: $12,453 (might be high depending on your employer)
* Other necessities: $15689
* Taxes: $12490
I assume that tax number is based in the actually cited $91k income, so to figure out how much more income we need for discretionary spending and savings we need to look at marginal tax rates. Looks like CA state income tax is 8 or 9.3% in the relevant income range (60k-500k for couples). Federal rates are 25-28% (75k-233k taxable income). Medicare is 1.45% until we get to 250k. Social Security is 6.2% until we hit the cap. So a marginal rate of 35% to 39% not counting Social Security.
Retirement savings should be ~15k here (15% of income is the common recommendation), and can often be done "pre-tax" (so only taxed by Social Security and Medicare); let's call this 16k gross income.
Are you saving for college for those two kids? Any other savings (e.g. for a house)? Any discretionary income at all? Multiply all those numbers by 1.6 or so to account for the taxes involved. Realistically I would expect this to come to another 20-30k easily for anything resembling a "middle class" lifestyle.
Looking at the numbers from your link, they have a much lower healthcare estimate (~6k/year), a somewhat higher housing estimate, correctly estimate childcare costs if only one parent works, but still have what look like ludicrously low numbers if both parents work. The "2 adults, both working, 2 children" number from your link is $83k and the difference between that and the $91k from the businessinsider article is all down to the health care number.
Anyway, the correct conclusion here is that in the Bay Area for a family of 4 you need an income in the 70-90k range (mostly depending on how much you pay for health insurance and whether you both work) just to barely scrape by, with no savings and no discretionary spending of any sort. At 150k you can probably do ok in terms of being able to save for retirement, college, maybe take some vacations every so often, etc. 200k should be reasonably comfortable if you have no pretensions to more than 2 bedrooms (notice that housing for 2 kids is claimed to cost the same as for one kid) and aren't trying to get into good school districts.
If you want good schools or more living space than a small 2-bedroom place way far from anywhere, the housing numbers cited above are almost certainly way too low...