No. Saying: "I'm a hotdog" is saying that you are one yourself. It's better if you say "That's my hotdog" Meaning that that hotdog belongs to me.

You can't say 'I'm hotdog'. 'Hotdog' is a singular, countable noun, so it needs to have an article or other determiner before it. But it's just about possible to say 'I'm the hotdog' - with the definite article - in a situation where a waiter or waitress brings various things to the table and wants to know who ordered what. It's a shortened version of 'I'm the person who ordered the hotdog'. Maybe people say it that way because it sounds funny. You don't have to say this, of course - it's clearer to say 'The hotdog's for me' or 'Mine's the hotdog'.