Slightly off topic now, but after nearly 40 years with an airline pilot (senior management for 20 years), these issues are part of a regular discussion in our home. 
I've also made approaches (in Australia) about mothers keeping their babies in slings for take-off and landings (no) - it seems unlikely that anyone would go through the very complicated process of getting approval for a particular sling (and costs for the necessary testing). 
As with restraints for children in cars, the USA has taken longer than some countries to legislate re attachment belts on planes. [see below press release] However, there will be some countries whose airlines still have no such legislation. In Australia, the airlines do provide attachment belts for infants (under 2) who are held in the lap, by an adult [still not as safe as the more expensive alternative, see below].

My concern with this story is that there might be a mix-up of 2 issues:
(i) safety on take-off [a baby's potential injuries on impact if not restrained; and the risk to others if not restrained ]
(ii) the mother's right to breastfeed wherever she chooses 

as someone pointed out, a flight attendant saying 'you can't do that' after the mother mentioned that the baby was feeding in the sling, may have meant 'you have to hold the baby in front of you' or wear a belt attachment for take-off.... [or have a more negative intent..]

we really don't know, we weren't there.. 
but approaches to the airline for clarification and questions about discrimination would be enlightening. I would do it if I knew the airline.
It sounds quite appalling.

Jan Cornfoot
Breastfeeding Advocate

 The United States FAA website has this press release below - FAA rules usually apply to all aircraft in US airspace. 

So, the FAA encourages use of a safety device on an aeroplane, but airlines can have their own (enhanced) rules about restraint.

Washington Headquarters Press Release 
For Immediate Release
Release No. AOC 26-06
September 6, 2006
Contact: Alison Duquette 
Phone: (202) 267-3883 

FAA Approves New Child Safety Device
Government Gives Parents More Options for Safe Air Travel with Children 


WASHINGTON, DC - Air travelers have a new option for securing their children on commercial flights now that the U.S. Department of Transportation's Federal Aviation Administration (FAA) has approved a new type of child safety device, the AmSafe Aviation CARES.

CARES uses an additional belt and shoulder harness that goes around the seat back and attaches to the passenger lap belt to provide restraint for the upper part of the body. It is designed for children weighing between 22 and 44 pounds. The device provides a smaller and lighter alternative to using forward-facing child safety seats. CARES is not approved for use in motor vehicles.

"We want to provide parents with options so they can make the right decision for their children when they travel by air," said FAA Administrator Marion C. Blakey. "We're encouraging child seat manufacturers to design new types of devices that meet the FAA's stringent standards."

Unlike hard-backed child safety seats that are approved for use in airplanes and motor vehicles, CARES is designed and tested specifically for safe use in airplanes only.  Previously, the FAA had allowed only airlines to provide these types of additional child safety devices, but no U.S. airlines presently provide them.

According to the FAA, the safest place for a child on an airplane is in an approved child safety device, not on a parents lap. The agency encourages but does not mandate the use of child safety devices on airplanes because of the increased safety risk to families who, if forced to purchase an extra airline ticket, might choose to drive. The risk to families is significantly greater on the roads than in airplanes, according to FAA and National Highway Traffic Safety Administration (NHTSA) statistics.

For additional information go to www.faa.gov/passengers/fly_children.

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