alphadin implements the norm DIN 1505-2, which is now superseded by (the slightly less terrible) (DIN) ISO 690. DIN 1505-2 uses semicolons (as well as en-dashes and slashes) with a leading and following space. In that regard the norm does not follow German tradition, which never places a space before a semicolon, but it also does not go full French, because it does not place spaces before colons.
That behaviour is hard-coded in alphadin.bst and has to be changed manually by modifying the .bst file.
Locate the style you want to change on your machine. You can find the file path by typing kpsewhich alphadin.bst into a terminal. Failing that get the files from CTAN: https://ctan.org/tex-archive/biblio/bibtex/contrib/german/din1505
Copy the file to a place where LaTeX can find it (https://texfaq.org/FAQ-inst-wlcf) – the directory of your current document will do just fine – and rename it. Many LaTeX files require you to rename the file in their license conditions. But even if the license is not clear on that or does not require you to rename a changed file, it is extremely good practice to rename changed files. Let's say the new name is alphadin-nospace.bst
Find all six occurrences of " ; " in the file and replace them with "; ". In my copy the relevant lines are
Ideally you place a short notice of the changes, your name and the date at the top of the file.
Save the modified and renamed file.
Use \bibliographystyle{alphadin-nospace} instead of \bibliographystyle{alphadin} in your document.
\RequirePackage{filecontents}
\begin{filecontents}{\jobname.bib}
@article{arbinger2006designing,
title = {Designing with an embedded soft-core processor},
author = {Arbinger, Don and Erdmann, Jeremy},
journal = {Embedded cracking the code to systems development},
year = {2006},
volume = {14},
pages = {76-89},
}
\end{filecontents}
\documentclass{article}
\bibliographystyle{alphadin-nosp}
\begin{document}
\cite{arbinger2006designing}
\bibliography{\jobname}
\end{document}

Since Mico gave the same answer just a minute and a half earlier, let me try to justify the presence of this answer with a bit of advertising for biblatex.
As mentioned above DIN 1505-2 has been superseded by ISO 690 and so it makes little sense to continue using alphadin or any of the other styles from the din1505 bundle for new documents. (Unless you really, really like the result the style gives you of course.)
A similar style can be achieved with biblatex, which is much more flexible when it comes to small changes like this. See bibtex vs. biber and biblatex vs. natbib, What to do to switch to biblatex?, Biblatex with Biber: Configuring my editor to avoid undefined citations and Guidelines for customizing biblatex styles for more introductory posts on biblatex.
\RequirePackage{filecontents}
\begin{filecontents}{\jobname.bib}
@article{arbinger2006designing,
title = {Designing with an embedded soft-core processor},
author = {Arbinger, Don and Erdmann, Jeremy},
journal = {Embedded cracking the code to systems development},
year = {2006},
volume = {14},
pages = {76-89},
}
\end{filecontents}
\documentclass[ngerman]{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{babel}
\usepackage{csquotes}
\usepackage[style=alphabetic]{biblatex}
\DeclareNameAlias{author}{sortname}
\DeclareNameAlias{editor}{sortname}
\DeclareNameAlias{translator}{sortname}
\DeclareNameAlias{sortname}{family-given}
\DeclareDelimFormat{multinamedelim}{\addsemicolon\space}
\DeclareDelimAlias{finalnamedelim}{multinamedelim}
\renewcommand*{\mkbibnamefamily}{\textsc}
\DeclareFieldFormat
[article,inbook,incollection,inproceedings,patent,thesis,unpublished]
{title}{#1\isdot}
\addbibresource{\jobname.bib}
\begin{document}
\cite{arbinger2006designing}
\printbibliography
\end{document}
