That does sound strange. " On the other hand, it might be acceptable to say "I am the cheeseburger" in this context: Waiter brings food. Waiter: Ok so here you go.
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Snappy"I am a hamburger" is a strange expression.The only time it would appropriate is, for example, when the server, bringing the sandwiches, might ask, "Now, who's the cheeseburger?", when everyone else had ordered just a plain burger.
I might truthfully say that - I have lived in the city of Hamburg for the last 30 years, so I'm more entitled than JFK to his famous "Ich bin ein Berliner"...
SnappyA Japanese site explains that it is acceptable
That's why. Non-native speakers often use English in very strange ways, including mistakes.
SnappyIt should be "I'd like a hamburger" if I want to order a hamburger.
That would be a better way of saying it.
Snappy"I am a hamburger"