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There are special macros for stuff like exp, sin, cos, etc... but not for the non-elementary special functions like F, J, I, K, E etc... although their symbold are quite standard throughout literature.

I know ISO/IEC 80000 standardized the use of mathematical constants as upright, along with greek letters and mathematical operators. I try to adhere to this, because frankly, it looks nicer overall.

Is there a similar standard for the math functions? Would it look weird if I made these upright as well?

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There is no standard, that covers all math functions. Some functions are written in a different way, but even than there are several ways to do this. For example the indicator function can be written with a

  • bold 1: 𝟏
  • double struck 1: 𝟙
  • bold, upright I: 𝐈
  • double struck, upright I: 𝕀
  • italic I: 𝐼

I don't know about ISO, but the German standard, DIN1302, defines the Heaviside function as either 𝐻 or ε and rect(x), tri(x), sinc(x) and inv(x) in upright math.

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