I am aware of people who call themselves vegetarians when they eat fish because there is a cultural classification of fish as distinct from meat, and vegetarian is commonly defined as one who refrains from eating meat rather than animals generally, so I at least understand why someone would consider themselves a fish-eating vegetarian.

But vegan is specifically defined as not using animal products, so a vegan diet must be devoid of animal products. I can understand someone might be vegan in principle and almost entirely in practice but not, for instance, meticulously check ingredients lists to see whether there is a dye made from crushed insects or will accept food prepared by someone who says there are no animal products in it without asking to check all the boxes and jars of any packaged ingredients that have been used, but it defies common sense that someone would claim to be, say, a cheese-eating vegan or a gelatin-eating vegan (although for the latter, if the person in question is simply eating a 99+% vegan diet and is not philosophically vegan, I can see why they would claim they are vegan so their friends know they won't eat a burger or an omelet even though including gelatin means their diet isn't technically vegan).