How to center the higher glyph(s) in a `\stackrel` or stretch the lower glyph?

Some fine tuning is necessary.

First, we have to remove the small space that TeX adds after superscripts, which is \scriptspace. Second, we can compensate the hole due to the conflicting shapes by doing \mathbb{L}^{\!1}. Finally, a thin space before the “L” helps in centering. The whole thing is set in \mathclap, so it won't count for the width of the big symbol.

\documentclass{article}
\usepackage{amsmath,mathtools,amssymb}

\newcommand{\Lsense}{%
  \overset{\mathclap{\,\mathbb{L}^{\!1}\kern-\scriptspace}}{=}%
}

\begin{document}

$A\Lsense B$

$A=B$

\end{document}

The distance from the left side of the “L” to the left side of “=” is not the same as the distance from the right side of the “1” to the right side of the “=”, but it's not of a concern, visually, because the exponent is set quite high. You may want to use \mspace{2mu} (or less) instead of \, (which would be \mspace{3mu}).

Here's a “boxed” version that shows the horizontal size is the same.

What about

\documentclass{article}
\usepackage{amsmath,amssymb}

\begin{document}

\newcommand{\Lsense}{\stackrel{\mathbb{L}^{\mkern-5mu 1}}{=}} %precise value up to you

$A\Lsense B$

\end{document}

Certainly, \mkern can be also in other places, e.g.,

\newcommand{\Lsense}{\stackrel{\mkern2mu\mathbb{L}^{\mkern-5mu 1}}{=}}

How about

\usepackage{mathtools,amssymb}
\newcommand*\Lsense{\overset{\scriptscriptstyle\mathbb{L}^{\mathrlap{\!1}}}{=}}

You can remove \scripscriptstyle if you think it's to small. The \mathrlap is key here, because makes the superscript take zero space.