You modified tauthesis.cls changing \ExecuteOptions{finnish,numeric} to \ExecuteOptions{finnish,apa}.
You could also add urldate=short when BiBLaTeX is required:
\RequirePackage
[backend=biber,
autolang=other,
urldate=short,
citestyle=\@citationstyle,
bibstyle=\@citationstyle]{biblatex}
After doing it, your citation will appear this way:
\begin{filecontents}[overwrite]{references.bib}
@online{test,
author = {An Author},
title = {This is the title},
url = {https://www.url.com/},
year = {2023},
urldate = {2023-04-15}
}
\end{filecontents}
\pdfminorversion=6
\documentclass{tauthesis}
\usepackage{amsmath, amssymb, amsthm}
\newcommand{\verbcommand}[1]{\texttt{\textbackslash #1}}
\makeglossaries
\addbibresource{references.bib}
\begin{document}
Test citation \cite{test}.
\printbibliography[heading=bibintoc]
\end{document}

EDIT:
Redefining urldate:
\begin{filecontents*}[overwrite]{references.bib}
@online{test,
author = {An Author},
title = {This is the title of some cool site},
url = {https://www.url.com/},
year = {2023},
urldate = {2023-04-15}
}
@online{test2,
author = {An Authorr},
title = {Why doesn't this also wrap incorrectly},
subtitle = {so weird, I cant replicate},
url = {https://www.url.com/this-a-long-url-yeah-yeah-yeah-yeah.html},
year = {2023},
urldate = {2023-04-15}
}
@online{SOAPvsREST,
author = {Alyssa Walker},
title = {SOAP vs REST API},
subtitle = {Difference Between Web Services},
url = {https://www.guru99.com/comparison-between-web-services.html},
year = {2023},
urldate = {2023-04-10}
}
\end{filecontents*}
\begin{filecontents}[overwrite]{\jobname.xmpdata}
\Title{test}
\Author{test}
\Language{fi}
\end{filecontents}
\pdfminorversion=6
\documentclass{tauthesis}
\usepackage{amsmath, amssymb, amsthm}
\usepackage{hyperref}
\hypersetup{hidelinks}
% float placement
\usepackage{float}
% svg support
\usepackage{svg}
% landscape support
\usepackage{lscape}
% long table support
\usepackage{longtable}
% checkmark and x mark support from pifont's dingbats
\usepackage{pifont}
\newcommand{\cmark}{\ding{51}}%
\newcommand{\xmark}{\ding{55}}%
%%%%% Your commands.
% added:
\usepackage{microtype}
% Print verbatim LaTeX commands
\newcommand{\verbcommand}[1]{\texttt{\textbackslash #1}}
\makeglossaries
% added:
\DefineBibliographyStrings{finnish}{%
urlseen = {Haettu\addspace},%
urlfrom = {osoitteesta\addspace},%
}
% added (thanks to moewe):
\DeclareFieldFormat{urldate}{\bibstring{urlseen}~#1\space\bibstring{urlfrom}}
\renewbibmacro*{url+urldate}{%
\usebibmacro{urldate}%
\setunit{\addspace}%
\usebibmacro{url}%
}
\DefineBibliographyExtras{finnish}{%
% d-m-y format for short dates
\protected\def\mkbibdateshort#1#2#3{%
\iffieldundef{#3}
{}
{\thefield{#3}%
\iffieldundef{#2}{}{.}}%
\iffieldundef{#2}
{}
{\thefield{#2}%
\iffieldundef{#1}{}{.}}%
\iffieldbibstring{#1}{\bibstring{\thefield{#1}}}{\mkyearzeros{\thefield{#1}}}}%
}
% added:
\renewcommand*{\finentrypunct}{}
\addbibresource{references.bib}
\begin{document}
\chapter{test}
\cite{test}
\cite{test2}
\cite{SOAPvsREST}
\printbibliography[heading=bibintoc]
\end{document}

For the text that goes out of margins, you can load microtype package.
To remove the dot at the end you can use:
\renewcommand*{\finentrypunct}{}
but it will remove the final dot for all the entries.
\ExecuteOptions{finnish,numeric}to\ExecuteOptions{finnish,apa}intauthesis.cls– 0x464e Apr 15 '23 at 22:41