Your informant is correct. Both Case and New Holland machines are built in the same plant. There is a lot more to the differences than just loader arms, however.
Case and New Holland loaders do share more drive components than they used to. But the differences are major. Sit in one brand and then another, you'd be hard pressed to identify things that are the same. Most of the common components are engine and hydraulics.
Rather than being siblings like Case and New Holland tractors, I'd say the skid steer loaders are more like distant cousins.
New Holland skid steer loaders used to be manufactured in Belleville, PA. Production of the New Holland skid steer loader was moved to the Wichita, KS plant a few years ago due to it being more modern and having the additional capacity.
If you ever saw the Belleville, PA plant you would understand the reasoning. It is small and broken up into many buildings. It is remarkable that New Holland churned out all the units that they did from that plant. Those New Holland employees must have been very efficient.