Absolutely false.
My milanese loop wasn't locking in place on my 42mm Sport Watch. Took it to the Genius Bar on the 10th, they verified the issue within a few minutes, determined that both the watch and band needed to be replaced. Picked up my replacement Watch and milanese loop this Monday, the 18th. Before calling me they checked to make sure the replacement band & Watch worked together properly.
Apple has said VERY clearly that they designed the bands to be interchangeable between all (aside from size) models of the Watch.
Galvanic corrosion IS a concern but not likely to be an issue thanks to the heavy coating on the different metals. Someone who changes bands multiple times a day, in a careless manner, actively scratching away at the coatings in key areas, might notice an issue but this will by no means be the norm.
OP, you just got bad info from an Apple Specialist who likely hadn't heard correctly.
edit:
Just to add, I asked for verification on one thing I wasn't clear on. With larger Apple devices (Macs, iPhones, etc) AppleCare covers the accessories that are purchased alongside the device. i.e. If you purchase an Airport Extreme with your iMac then the Airport is covered under the iMac's AppleCare for three years as well. I asked if this was true for the Apple Watch and the different Apple Bands. The Genius working with me was pretty sure it did but called the lead Genius (essentially the defining voice of AppleCare in retail stores) over to ask for sure. The lead Genius commented that he had checked on that just a few days earlier and that it IS true. So, if your Link Bracelet has an issue in a year and a half, and you bought AppleCare with your Apple Watch Sport, then it will be covered under that same warranty.