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If CG stands for something (like it’s the initials of their name), then it is fair to be both capitalised. For example, if my middle name was Jane (it’s not), then I might refer to myself as MJ, but I may also use lower case, mj (as might be the case in my email address, where all characters are generally written in lower case. I personally would never write it Mj, unless perhaps it was the start of a sentence, where the first letter defaults to caps.🤷‍♀️

Both proper nouns (thereby names) and abbreviations/acronyms are expected (for the sake of coherency in interpretation) to have consistent capitalization. To my understanding, nicknames follow the rules of standard names without any significant alterations.

Lowercase lettering in an abbreviation indicates the letters are secondary* to the preceeding capital letter, while the fact that lowercase lettering in an acronym indicates a preposition, definite, or indefinite article. This differs from other words in English, where capitalization is mostly stylistic and for formatting/ease of reading, and makes capitalization rather relevant for those words (and that in addition to capitalization helping make it clear that such words are abbreviations or acroynms to begin with).

*(For example, were you to use ForF to stand for Forest Fire, the lowercase "or" indicates the or is secondary to the first F, while the capitalization of the second F clarifies that the letter represents the start of a second word; This approach is also consistently matched to formatting practices within most scholastic/technical fields, such as with chemical formulas)

Proper nouns mostly do it to make it clear when you're referring to a name rather than a word [eg, "The Bunnies" could make it clear you're refererring to a sports team rather than actual rabbits]. Capitalization in names also helps distinguish titles (eg, nee and von and the like are always lowercased, and thus easy to distinguish as titles or associative elements rather than as coincidentally identical parts of the name itself).

There's also a loose expectation of it indicating an emphasis of pronounciation or of seperation of meaning within compounded names, but that's generally meaninglessly inconsistent [perhaps due to the international, and thus cross-language, influence of names] (just look at the Dutch surname Van der Meer, which has the variant forms of Vandermeer, Vander Meer, and VanderMeer, evidencing that there's no real consistency of approach to such matters).

The weirdest part about capitalization in personal names is the social/vanity aspect. It can be considered rather insulting to miscapitalize an individual's name (well, that was something my elementary school self found rather absurd for my teachers to emphasize so heavily but, in practice, people do tend to get prickly over their names).

Finally, in terms of names in general, consistent capitalization is expected for the sake of clarity. Any variation in capitalization implies a secondary individual as being referenced, though such an implication can of course be dismissed in any circumstance where one can easily intuit the variation as simply being a typo. In situations where you have two John Smiths in your office, however, the deviation in capitalization may be an intentional (albeit atypical) method of differentiating the two. (Lowercase smith, you know which one you are..)

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Well, again, none of the above is directly relevant to this particular matter. Here, the issue wasn't in miscapitalizing cg's name, but in the fact that both cg and SG have equal similarity with CG, making easy interpretation of that deviation in expected capitalization/spelling a bit hard to recognize. As cg is representative of SG (as he has full control over the site), the distinction is mostly irrelevant, but in other contexts that deviation could be legitimately confusing. This confusion is of course emphasized by the fact that Cg is the expected natural capitalization of cg, emphasizing that CG could simply be a typo of SG.

Likewise, while CG may be the formatting present on Patreon, it is still considered proper to translate names to their localized forms where localization exists*. As such, the patreon formatting cannot be considered relevant on this site, which already has an expected formatting for the name [as that is considered to take priority, due to localization].

*(Of course, this forced localization may at times seem problematic due to inconsiderate renaming, such as being required to translate Deutschland into Germany for English, despite the fact that German is a term that has never been used by the natives of the region in question; Ergo, making such relabelings of outside regions rather insensitive. After all, would it be proper to refer to Africa as "Blacksville", Eskimos as "Snow People", or part of Europe as "The Barbarian Lands", just because that was the expected localized form? As noted earlier in the post, proper presentation of names can be a sensitive thing, so even less intuitively offensive relabels can lead to discontent.
Conversely, the merits of adhering to the perspectives associated with proper nouns/names is highlighted in programming and other technical fields, where minor distinctions in presentation can be significant in their relevancy, such as how a change of capitalization can directly change which variable is being referenced by a programming script.)