Clarity is the point of typography.
Personally, I like the literal quoting scheme, where quoted sections are punctuated as normal, excepting only the double and singles toggled for nesting. It's rather common to collapse final punctuation when the outer sentence and quoted sentence agree.
E.g. "I love your work, but calling you 'the artist' just doesn't seem to be cutting it anymore. What shall I call you?"
"OK. Call me 'George'."
As to why it seems different, I suggest it's because "Call me George" fits a pattern of use for "call" that isn't actually spoken, so would not be quoted. ("Call me later." "Call me off.")
Alternately, some schemes would skip quoting it simply because it's not, literally, a quote: It's an imperative to speak.
I would quote the called-out term, "George", since that clarifies the use of the word "call" in the sentence. (used as 'use the following in speech' rather than as a shorthand for 'describe as')
(Ed. It bears noting that simply capitalizing the word "George" calls it out as a proper noun, which grants it special distinctions.)
The typographic rules you should follow should be clear but also fit any rules established by a relevant authority; (publisher/teacher) Do realize that there are multiple competing conventions.