"我姓王。"
Translation:My last name is Wang.
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姓 (xìng) is not a noun for "surname/last name", it is actually a verb meaning "to be surnamed". The sentence is literally "I am surnamed Wang." Here, it is easy to see there is no possessive in the sentence. But in English, we don't typically use the verb "to be surnamed"; our loose English translation presents with a possessive instead (MY last name is Wang).
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This may be true for casual conversation and the statement above but in polite or formal company we do ask "你贵姓?” meaning "(and) your surname is?" 贵 being an adjective rather than an adverb.
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It is just an example for the sentences. People have their own names other than "wang"
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"I am Wang" implies that the person's first name is Wang. 姓 refers specifically to a surname.
There are several colloquialisms that translate and Duolingo can only mark it right if they considered it ahead of time. They also phrase the lesson in the same manner several times giving you all the clues you need if getting each answer correct each time is important to you. I am trying to learn a language not get the high score.