The standard definition of \enoteformat is
\def\enoteformat{\rightskip\z@ \leftskip\z@ \parindent=1.8em
\leavevmode\llap{\makeenmark}}
Taking this as a model, here's what you may want:
\documentclass{article}
\usepackage{endnotes}
\usepackage{lipsum}
\renewcommand{\enotesize}{\normalsize}
\renewcommand\enoteformat{%
\raggedright
\leftskip=1.8em
\makebox[0pt][r]{\theenmark. \rule{0pt}{\dimexpr\ht\strutbox+\baselineskip}}%
}
\begin{document}
\endnote{\lipsum[1]}\endnote{\lipsum[2]}
\theendnotes
\end{document}

The same, but perhaps better and surely easier, with the enotez package:
\documentclass{article}
\usepackage{enotez}
\usepackage{lipsum}
\DeclareInstance{enotez-list}{sverre}{paragraph}
{
heading=\section*{#1},
notes-sep=\baselineskip,
format=\normalsize\normalfont\raggedright\leftskip1.8em,
number=\makebox[0pt][r]{#1.\ }\ignorespaces,
}
\begin{document}
\endnote{\lipsum[1]}\endnote{\lipsum[2]}
\printendnotes[sverre]
\end{document}
If you want the indentation to be as wide as the parindent and the numbers at the left margin, then do
\documentclass{article}
\usepackage{showframe} % just for the example
\usepackage{enotez}
\usepackage{lipsum}
\newlength{\normalparindent}
\AtBeginDocument{\setlength{\normalparindent}{\parindent}}
\DeclareInstance{enotez-list}{sverre}{paragraph}
{
heading=\section*{#1},
notes-sep=\baselineskip,
format=\normalsize\normalfont\raggedright\leftskip\normalparindent,
number=\makebox[0pt][r]{\makebox[\normalparindent][l]{#1.}}\ignorespaces,
}
\begin{document}
Something\endnote{\lipsum[1]} to show the parindent\endnote{\lipsum[2]}
\printendnotes[sverre]
\end{document}
I added showframe just to make sure the specifications are respected

Notes
The parameter \leftskip is the distance from the global left margin to the actual left margin of the text. The \raggedright command sets it to zero, so we override this later, and sets \rightskip to a stretchable space; it also sets \parindent to zero, so I defined \normalparindent to remember its value. You may want to try also \RaggedRight from the ragged2e package that allows for (rare) hyphenation in order to reduce the raggedness. With the double makebox trick we set a zero width box with its contents sticking to its left, containing a box \normalparindent wide, with the contents shoved to the left.