Equalizer coefficients and channel coefficients
With a blind equalization technique like the constant modulus algorithm (which is often implemented using a least mean squares (LMS) filter as you indicated), you aren't directly estimating the channel impulse response itself. Instead, the signal model is like this:


The receiver observes the following signal:
$$
x[k] = s[k] * c[k] + n[k]
$$

where:


$s[k]$ is the underlying desired signal,
$c[k]$ is the channel's impulse response, and
$n[k]$ is a noise process introduced by the channel (typically assumed to be additive white Gaussian noise)

A blind equalizer operates by taking advantage of some constraint on the properties of the desired signal $s[k]$. For instance, if it is constant envelope, like phase-shift keying, then the constraint could be:
$$
|s[k]| = 1
$$
The CMA-LMS blind equalizer operates in two steps. First, it uses the current length-$L$ vector of LMS filter taps (where $L$ is a design parameter of the filter) $\mathbf{w}[k]$ to calculate its output:
$$
y[k] = \mathbf{w}[k]^H\mathbf{x}[k]
$$

where $\mathbf{x}[k]$ is a vector of the last $L$ input samples to the filter. Next, it uses the calculated output to update its set of taps for the next iteration:
$$
\mathbf{w}[k+1] = \mathbf{w}[k] - \mu \mathbf{x}[k](|y[k]|^2 - 1)y^*[k]
$$

When compared to the typical LMS filter structure, one can identify the error signal $e[k]$ as:

$$
e[k] = (|y[k]|^2 - 1)y^*[k]
$$

You can see here how the error captures how far the filter output signal $y[k]$ deviates from the expected constant envelope of the desired signal $s[k]$.


Looking at this structure, you can see that at no point is the equalizer trying to directly estimate the channel's impulse response. Instead, it's more "given the knowledge that I have of the expected signal's structure, how do I need to adjust my filter taps to enhance that structure in the filter output?" So it's more of an inverse problem; the tap vector will tend to more closely resemble the inverse of the channel impulse response. Blind equalization techniques like this can do a surprisingly good job at doing so (even more surprisingly, techniques like CMA can even work for signal types that aren't constant envelope, like QAM).