IEEE 754 NaN’s are arguably defined as different things, but that’s not the only system to use floating point Math and NaN.
Mathematically different infinities are not equivalent because infinity is not a number. How you represent that is dependent on the specific system involved.
Since math is one of the few things I fell confident talking about:
It is true that different infinities exists[0][1] and there are whole areas of logic examining them.
It is also true that _number_ without any extra qualifier generally means the Real numbers (R) or Complex numbers (C) and those domain do not define infinity as a number, but even then there are only a few good ways to add infinity into each number system:
In R generally you either add a projective point of infinity ∞ [2] that makes geometry sometimes nices or two signed infinities (-∞ and +∞) that make calculus nicer (especially limits and integration)
In C it is often simpler as typically you want to treat them as a sphere and so add an extra point so that the inversion f(x) = 1/x is a well-behaved function. In this domain you often end up working with holomorphic functions[3] and then there is not really an intrinsic difference between a function like f(x) = 1/x and g(x) = x they simply have both a _pole_ f at 0 and g at infinity.
If you want to get trippy even integers can have unusual definitions [4] and then there is always one of my favorite topic in math: surreal numbers [5] (for which I recommend both [6] and [7]) a field where √∞ < ∞/2 < ∞ - 1 < ∞ < ∞ + 1 and is perfectly well defined (but still 0/0 doesn't have any meaning in any of these theories, that is a though nut to crack)
Mathematically different infinities are not equivalent because infinity is not a number. How you represent that is dependent on the specific system involved.