Fourier transform of arbitrary functions from R to C works fine with whatever units.
Fourier series of functions from R/Z to C (i.e., functions of period 1; i.e., functions f from R to C such that f(x + 1) = f(x) for all x) uses specifically cos(2πNx) and sin(2πNx) for integer N; you can't replace 2π here by any other value (since then you don't get the set of cosines and sines of period 1).
That's the special connection between 2π and periodic functions: every function of period 1 decomposes into the sum of a series of functions whose 2nd derivatives are square numbers * -(2π)^2 * themselves. [Or, more cleanly, into the sum of a series of functions whose 1st derivatives are integers * 2πi * themselves]. Again, you cannot replace 2π with any other value in the statements of this paragraph (well, except its negation...).
> functions f from R to C such that f(x + 1) = f(x) for all x) uses specifically cos(2πNx) and sin(2πNx) for integer N
Define cos'(x) = cos(2πx/A) and sin'(x) = sin(2πx/A) (such functions could be defined without referring to π in their definition)
Now f can be expressed as a weighted sum of cos'(ANx) and sin'(ANx) for integer N.
> every function of period 1 decomposes into the sum of a series of functions whose 2nd derivatives are square numbers * -(2π)^2 * themselves
You are correct that the scaled sin and cos functions I mention above still have the 2nd derivative equal to -(2π)^2 * themselves.
I found that a weak argument that pi is somehow inherent in periodic functions though.
First, this property of sin and cos is not necessary for or directly connected to the Fourier transform as far as I can see.
Second, there are infinitely many families of orthogonal functions that can be used to decompose periodic functions in the same way sin and cos can be used and do not fit this rule of the 2nd derivative. Consider a family of square wave functions for instance.
Sure, let's say cos'(x) is the cosine of the angle corresponding to x complete revolutions (thus, cos' is of period 1), and similarly for sin'. Then every function of period 1 is a unique linear combination of cos'(Nx) and sin'(Nx) for integer N. We can say this. This is essentially the same as what we were saying already. My point isn't that π is used in the words we say when defining cos' and sin', but that it is part of their properties, making its presence known once we consider derivatives: the derivative of cos' is -2π * sin', and the derivative of sin' is 2π * cos'. [Er, the primes here don't denote derivatives, but just our newly scaled trig functions, of course]
π (2π, etc.) truly is special for the Fourier transform: it is inherent in decomposing a periodic function into a sum of exponentials (and cosine and sine are just even and odd components of exponential functions). An exponential function from R to C is of period 1 if and only if the natural logarithm of the base of the exponential is an integer multiple of 2πi (i.e., just in case the function's derivative is an integer multiple of 2πi times itself).
It's true that there are other (non-exponential) families of orthogonal functions that can be used for decompositions, but the fact that the Fourier transform is specifically in terms of a family of exponential functions is of some significance (for example, it means multiplication on one side of the transform corresponds to convolution on the other side; it's also what makes the Fourier transform and inverse Fourier transform essentially the same operation).
Tl;dr: Periodic functions can be represented in various ways, but the Fourier series decomposition is a particularly useful representation because it "diagonalizes" differentiation (which is to say, it "diagonalizes" translation by tiny (and consequently also by arbitrarily sized) amounts). And what we discover in diagonalizing the differentiation operator on functions of a fixed period is that its eigenvalues are precisely the integer multiples of 2πi divided by the period. This is a fundamental connection between π and the differential(/integral/etc.) structure of periodic functions.
Fourier series of functions from R/Z to C (i.e., functions of period 1; i.e., functions f from R to C such that f(x + 1) = f(x) for all x) uses specifically cos(2πNx) and sin(2πNx) for integer N; you can't replace 2π here by any other value (since then you don't get the set of cosines and sines of period 1).
That's the special connection between 2π and periodic functions: every function of period 1 decomposes into the sum of a series of functions whose 2nd derivatives are square numbers * -(2π)^2 * themselves. [Or, more cleanly, into the sum of a series of functions whose 1st derivatives are integers * 2πi * themselves]. Again, you cannot replace 2π with any other value in the statements of this paragraph (well, except its negation...).