Are all least square filters adaptive?
TL;DR: No, they are not necessarily the same.



Gory Details

Least squares is just an optimization technique. It is used in a variety of ways.

For filter design it is used to select that realizable filter $H_r(e^{j\omega})$ that most closely matches, in the least squares sense, the ideal required filter response $H_i(e^{j\omega})$:
$$
H_r(e^{j\omega}) = \arg \min \parallel H_r - H_i \parallel_2  
$$
where $\parallel \cdot \parallel_2$ is the 2-norm or least-squares norm.

This sort of filter $H_r$ is not adaptive. That is, it doesn't change once it has been designed.

Adaptive filters may also use the least squares criterion, but in a different way: as part of the adaptation step.

Adaptive filters start off with initial filter coefficients $\vec{w}_o[0]$ and then use an update:
$$
\vec{w}_o[n] =\vec{w}_o[n-1] + \mu g[n-1]
$$
where $\mu$ is the step-size and $g$ is the gradient of the least squared error  surface in the direction of the minimum (from our current "location" of $\vec{w}_o[n-1]$).

Here, $g$ is determined by our error criterion: least squares. This means: 
$$
\parallel \vec{w}_{\tt opt} - \vec{w}_o \parallel_2
$$
where $ \vec{w}_{\tt opt}$ is the unknown optimal (minimizing) solution.