You are not unique in this sort of frustration; many women protest against their husbands calling them "my woman." Most people do like to feel as though they are not a "possession."
On the other token, them referring to someone as "my aspie" or "my _____" is usually a term of endearment; the willingness to "claim" someone as one's own, to be associated with that person and see them as a part of themselves, indicates that the person cares strongly about the person in question. NT partners often use terms such as "my sweetie," "my geek," "my princess," "my monkey," "my girl/boy friend," "my pumpkin," "my baby," and so on when referring to one another. The term "my" does not mean "I am in possession of this..." as much as "I am associated with..."
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I am not a textbook case of any particular disorder; I am an abstract, poetic portrayal of neurovariance with which much artistic license was taken.