Typing Amharic inside a math equation?

The main problem is that ethiop makes _ active. Use \sb instead in math. While ^ works, I used \sp for symmetry.

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

\begin{document}

\[
\int\sb{\textup{'alamenahe}}\sp{\textup{'abAs}} \textup{'abAs}=\text{'abAs}
\]

\end{document}

Better using \textup instead of \mbox, so that the font size is reduced in subscripts and superscripts.

Here is another solution using Unicode in LuaLaTeX:

\documentclass{article}
\tracinglostchars=2
\usepackage[nil]{babel}
\babelprovide[import,main,onchar=ids fonts]{amharic}
\babelprovide[onchar=ids fonts]{english}
\usepackage{unicode-math}

\defaultfontfeatures{ Scale=MatchLowercase,
                      Ligatures=TeX,
                      Renderer=Harfbuzz }
\babelfont{rm}
          {STIX Two Text}
% Noto fonts vailable at: https://www.google.com/get/noto/#serif-ethi
\babelfont[amharic]{rm}
          {Noto Serif Ethiopic}
% Set up sans-serif and monospace fonts here, if needed.
\setmathfont{STIX Two Math}

\babeltags{amharic=amharic}
\babeltags{english=english}

\newcommand\matham[1]{\mathop{\text{\rmfamily\upshape\textamharic{#1}}}}
\newcommand\mathbfam[1]{\mathop{\text{\rmfamily\bfseries\upshape\textamharic{#1}}}}

\begin{document}
\[ \int_{\matham{አለምነህ}}^{\matham{አባስ}} = \matham{አባስ}
\]
\end{document}

[

From what little I can tell, this sample is nonsense, but it will serve as an example of how to typeset Ethiopic words in math mode. I added two new commands, \mathbfam, which typesets in the bold upright Ethiopic text font, and \matham, which typesets in the same weight as the surrounding text and will therefore appear in regular weight most of the time, but bold if used in a bold header.

You appear to want to use whole words, rather than individual letters or syllables. Both commands therefore put space around their arguments, like the operators log, sin or max. This might not be convenient if you want to write \matham{አለምነህ}{} \cdot {\matham{አባስ}. To run individual letters together like ax + by, change \mathop to \mathord.