For example can you use environment center and add inside it \captionof (needs to load package caption). You need an environment to be able to use \captionof and in your case you can use environment center to get the table centered ...
See the following code
\begin{filecontents*}{\jobname.csv}
Project, Size
A_b,-0.1237466
b,0.04632737
\end{filecontents*}
\documentclass{article}
\usepackage{csvsimple}
\usepackage{caption} % <================================= for \captionof
\begin{document}
\listoftables
\begin{center} % <======================================================
\captionof{table}{Test}\label{tab:test} % <=============================
\csvreader[%
tabular=l|r,
respect underscore=true,
table head=\hline Project & Size \\\hline\hline,
late after line=\\\hline
]%
{\jobname.csv}{}%
{\csvcoli & \csvcolii}%
\end{center} % <========================================================
As you can see in table~\ref{tab:test} \dots
\end{document}
and its result if you compile it with pdflatex:

As you can see, the _ is not displayed correct in the table. Now add the line
\usepackage[T1]{fontenc} % <======================== needed for pdflatex
to the preamble and you get well printed _ in the table. You need [T1]{fontenc} for correct printed document with pdflatex. Then you also get the _ printed with environment table.
Please see the following code
\begin{filecontents*}{\jobname.csv}
Project, Size
A_b,-0.1237466
b,0.04632737
\end{filecontents*}
\documentclass{article}
\usepackage[T1]{fontenc} % <======================== needed for pdflatex
\usepackage{csvsimple}
\usepackage{caption} % <================================= for \captionof
\begin{document}
\listoftables
\begin{center} % <======================================================
\captionof{table}{Test}\label{tab:test} % <=============================
\csvreader[%
tabular=l|r,
respect underscore=true,
table head=\hline Project & Size \\\hline\hline,
late after line=\\\hline
]%
{\jobname.csv}{}%
{\csvcoli & \csvcolii}%
\end{center} % <========================================================
As you can see in table~\ref{tab:test} \dots
\begin{table}[hb] % <===================================================
\caption{Test1}\label{tab:test1} % <====================================
\centering % <==========================================================
\csvreader[%
tabular=l|r,
respect underscore=true,
table head=\hline Project & Size \\\hline\hline,
late after line=\\\hline
]%
{\jobname.csv}{}%
{\csvcoli & \csvcolii}%
\end{table} % <========================================================
As you can see in table~\ref{tab:test1} \dots
\end{document}
and its result:

If you compile with lualatex or xelatex you do not need to call package fontenc. So comment the line \usepackage[T1]{fontenc} in the mwe above and compile with lualatex then you get the similar result:
