"Brewing sugar" could refer to corn sugar, as some brewers believe it is superior to cane, white or table sugar. Or like @Rob said, it could simply refer to any type of sugar. In my opinion, it would not refer to honey, but I don't know that.
I won't say corn sugar is superior, but it is different. I've been brewing hard seltzers lately and in an article I read, someone was discussing which sugars to use and said she had taken to using half and half in her homebrewed seltzers. I've taken up that practice with both my seltzers and my beers. When I add sugar, I add half cane and half corn sugar.
Simple sugar (cane, corn, brown, etc.) is very digestible for the yeast, so I usually use at least a little just to get things off to a good start. But I also like my beers a little on the dryer side than many commercial examples. A typical recipe for me might be 600 grams of malt and 100 grams of sugar in 4 liters. So my ratio is closer to 6:1 rather than the 3:1 your recipe calls for. I agree that that much sugar might cause the beer to taste boozy or vinegary.
In your case, if you don't add any sugar at all, your beer might be a little "malty". I think anything in the 300 - 500 gram range will be just fine.