Is there a way to make cleveref distinguish two environments with the same counter?

You can allocate lem and theo to the same count register (with care:-)

\documentclass{article}
\usepackage[colorlinks, linkcolor=blue]{hyperref} 
\usepackage[noabbrev, capitalise]{cleveref}
\usepackage{tikz}

%new theorem environment
\newcounter{theo}[section]\setcounter{theo}{0}
\renewcommand{\thetheo}{\arabic{section}.\arabic{theo}}
\newenvironment{theo}{%
\refstepcounter{theo}%
\tikz[baseline=(current bounding box.east),outer sep=0pt]
\node[anchor=east,rectangle,fill=green!20]
{\strut \textbf{Theorem~\thetheo.}};
\newline%BADNESS 10000!!!!!
}

\makeatletter
\let\c@lem\c@theo
%now def not \let so it picks up current value
\def\p@lem{\p@theo}
\def\thelem{\thetheo}
\makeatother
\crefname{theo}{Theorem}{Theorems}
\crefname{lem}{Lemma}{Lemmas}

%new lemma environment
\newenvironment{lem}{%
\refstepcounter{lem}%
\tikz[baseline=(current bounding box.east),outer sep=0pt]
\node[anchor=east,rectangle,fill=blue!20]
{\strut \textbf{Lemma~\thetheo.}};
\newline%BADNESS 10000!!!!!
}

\begin{document}
\begin{theo}\label{thrm}
Just some text.
\end{theo}

\begin{lem}\label{lm1}
Just some more text.
\end{lem}

\begin{lem}\label{lm2}
Just some more text.
\end{lem}

\noindent
\cref{thrm}\\ %give Theorem 0.1
\cref{lm1}\\ %give Lemma 0.2
\cref{lm2}\\ %give Lemma 0.3
\end{document}

No need for any programming contortions -- just load either the amsthm or the ntheorem package before both hyperref and cleveref, and then define the theorem-like environments the usual way. In particular, it's perfectly ok for several theorem-like environments to share the same counter (theo in the following code):

\documentclass{article}

\usepackage{amsthm} %or: \usepackage{ntheorem}
\usepackage[colorlinks, linkcolor=blue]{hyperref} 
\usepackage[noabbrev, capitalise]{cleveref}

% two new theorem-like environments
\newtheorem{theo}{Theorem}[section] % subordinate 'theo' cntr to 'section' cntr
\newtheorem{lem}[theo]{Lemma} % make 'lem' and 'theo' share same cntr

\crefname{theo}{Theorem}{Theorems}
\crefname{lem}{Lemma}{Lemmas}

\begin{document}
\setcounter{section}{2} % just for this example

\begin{theo}\label{thrm}Just some text.\end{theo}
\begin{lem}\label{lm1}Just some more text.\end{lem}
\begin{lem}\label{lm2}Still more text.\end{lem}

\cref{thrm} \dots

\cref{lm1,lm2} \dots
\end{document}