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When someone writes ''let $\mathcal{D}$ be a subcategory of a category $\mathcal{C}$'', is it possible that $\mathcal{C}=\mathcal{D}$? In other words, is a category a subcategory of itself?

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In the normal fashion of category theory it is best to think of the arrows between things. In this case, subcategory of $\mathcal{C}$ is a faithful functor into $\mathcal{C}$ that's injective on objects, i.e. a monomorphism in $\mathbf{Cat}$. If you wanted to be more strict, you could require it to be an inclusion. Either way, in this case $\mathcal{C}$ is a subcategory of itself via the identity functor which is trivially faithful and injective/an inclusion.

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    I think that many authors refer to a faithful functor that is the identity on objects as a "wide" subcategory. – 2017-01-30
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    This is an example of a general usage in math. A subset of a set $A$ is allowed to be $A$ itself, a group is a subset of itself, a topological space is a subspace of itself, and so on. – 2017-01-30
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    @KyleFerendo I usually interpret "identity on objects" to mean that if $F$ is a functor, then $FX = X$ on objects but that this still allows the domain of the object part of $F$ to be a subclass of codomain. However, I can't find an example where the term is used for a functor between categories with different classes of objects, so I'll change to "inclusion". – 2017-01-30
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    @DerekElkins Ah, I see. As I think you realize, I misinterpreted your phrase to mean a functor $F:C\to D$ such that $C$ and $D$ had the same class of objects, so that when restricted to objects, $F$ was the identity. I see now that that is not quite what you meant. – 2017-01-31