A wave's speed $c$, frequency $f$, and length $\lambda$ are related by
$$ c = \lambda f $$
A standing wave occurs in a system with some length scale $L$, and generally fixes the wavelength to $\lambda = aL$, where $a$ is some rational number. For instance in a wave on a string the allowed wavelengths are $\lambda = 2L/n$ for positive integer $n$; for fluid in a half-open pipe the allowed wavelengths are $\lambda = 4L/m$ for odd positive integer $m$. Most introductory texts get lost in computational details and don't sufficiently emphasize the fundamental result that, for a standing wave, $\lambda/L$ is some constant.
As the link explains, the "hot chocolate effect" happens when many small air bubbles are introduced into a liquid, because the air bubbles dramatically change the fluid's compressibility without much effect on its density. That changes the wave's speed, which depends on both density and compressibility. Your question is which part of the wave's properties remain constant: it's the wavelength, which is related to the height of the liquid column in the cup.