1080p is fairly new. A lot of TVs can't support it. It wasn't so long ago that 1080i was the highest you could go. And not so long before that, 720p was as high as you could go. Labeling something and then expecting that label to stick with changing technology is a fools errand. Just because a new format comes out won't make the older "full HD" suddenly better. If his camera only supports 1080i, and lists that as "full HD", then who are we to argue?
And 1080i on a progressive display screen can be just as good as 1080p is. It only really makes a difference when the display is actually interlaced, or so I've found.