This is misinformation that unfortunately has become quite widely spread.
Private employers generally are required to pre-fund their pension obligations: https://en.wikipedia.org/wiki/Employee_Retirement_Income_Sec.... FedEx and UPS both have around a 90% funded pension fund: https://www.investmentnews.com/article/20180509/FREE/1805099...; https://www.pionline.com/article/20190221/ONLINE/190229950/u....
What the 2006 postal reform legislation did is require the USPS to pre-fund its retiree health benefits as well. While it's technically true that private employers don't have that same requirement, that is only because their employees are eligible for Medicare. Like other private employers, FedEx and UPS, and their employees, pay the employer/employee Medicare taxes. Postal employees, by contrast, do not pay into Medicare and are not eligible for Medicare benefits.
There is some crazy idea out there that Congress somehow required the USPS to fund its employee health benefits to make government services look bad. That is complete nonsense. When the reforms were enacted, the postal service had a $75 billion unfunded liability for health benefits: https://taxfoundation.org/primer-postal-service-retiree-heal.... Congress saw the impending disaster, and (pretty much unanimously) voted to impose the funding requirement.