Nope, these are based mostly on surveys by private companies used for marketing purposes, typically credit cards
“Paycheck to paycheck” is not a well defined term, and most rely simply on the participants’ interpreted meaning.
This is why you often see a large percentage of those living “paycheck to paycheck” are making a middle-class income
For instance, the commonly cited Lending Club surveys usually find over 30% of those making over 200k a year are living paycheck to paycheck

The Bank of America has lower but similar numbers for households with over $150k in income

Federal Reserve data provides a better view of Americans’ finances
https://thedispatch.com/article/affordability-crisis-healthcare-housing-childcare/
Not really, 63% would cover a $400 emergency expense completely using cash or its equivalent (Figure 21, page 36)
Of the 37% who wouldn’t cover it with cash, a plurality (16%) would put it on a credit card, which is not that bad imo (table 16, page 37)

48% of Americans could cover a $2,000 or more emergency expense with only savings (table 17, page 37)
Nope, this figure is about “retirement accounts”. It doesn’t include pensions and savings outside non-tax advantaged accounts.
Federal Reserve data reports 86% of Americans aged 55-64+ had retirement savings.
Many of those with no savings earn below $25k a year. This is self reported pre-tax and transfer income, so many of those with no personal savings receive government transfers to partially fill the gap

Basically, everyone gets Social Security!
The old days of employer-provided pensions weren’t exactly great either. According to the 1971 the Senate Labor Committee:
When the median for normal retirement of $99 a month [$832 in $2025] is added to the median Social Security retirement benefit of $129 a month, the total, $228 a month, is less than the $241 minimum monthly income required to sustain a retired urban couple, as reported by the Bureau of Labor Statistics in January, 1970.