Multiple counters in the same list

Mark up with \itema and \itemb:

\documentclass{article}
\usepackage{enumitem}

\newenvironment{enumerateab}[1][]
 {\enumerate[label=\theEABII,before=\setcounter{EABI}{0},#1]}
 {\endenumerate}
\newcounter{EABI}
\newcounter{EABII}[EABI]
\makeatletter
\renewcommand{\theEABII}{\arabic{EABI}\@alph{\numexpr\c@EABII+1}}
\makeatother
\newcommand\itema{\stepcounter{EABI}\item}
\newcommand\itemb{\stepcounter{EABII}\item}

\begin{document}

\begin{enumerateab}
\itema item 1a
\itemb item 1b
\itema\label{foo} item 2a
\itemb item 2b
\end{enumerateab}

Item \ref{foo} is nice.

\end{document}

Alternative, if you always alternate “a” and “b”:

\documentclass{article}
\usepackage{enumitem}

\newenvironment{enumerateab}[1][]
 {\enumerate[label=\theEABII,before=\setcounter{EABI}{0}\itematrue,#1]%
  \let\latexitem\item\let\item\itemab}
 {\endenumerate}
\newif\ifitema
\newcounter{EABI}
\newcounter{EABII}[EABI]
\makeatletter
\renewcommand{\theEABII}{\arabic{EABI}\@alph{\numexpr\c@EABII+1}}
\makeatother

\newcommand\itemab{%
  \ifitema
    \stepcounter{EABI}\itemafalse\latexitem
  \else
    \stepcounter{EABII}\itematrue\latexitem
  \fi
}

\begin{document}

\begin{enumerateab}
\item item 1a
\item item 1b

\item\label{foo} item 2a
\item item 2b
\end{enumerateab}

Item \ref{foo} is nice.

\end{document}

Define your own representation

\documentclass{article}
\usepackage{enumitem}
\usepackage{xparse}
\ExplSyntaxOn\makeatletter
\NewDocumentCommand\arabicalph{ m }
{ \int_if_odd:nTF { \the\value{#1}} 
 { \int_eval:n {(\the\value{#1} + 1)/2 }a}
 { \int_eval:n {\the\value{#1} / 2 } b}}
\ExplSyntaxOff

\begin{document}


\begin{enumerate}[label=\arabicalph{enumi}]

        \item item 1a
        \item item 1b

        \item item 2a
        \item item 2b

\end{enumerate}
\end{document}

Remark: The syntax label= \arabicalph* doesn't work here, but it is now to late to try to find out why ...

Here is an implementation using an additional counter pairitem, and setting the representation of the item numbering manually:

\documentclass{article}

\newcounter{pairitem}
\newenvironment{pairenumerate}
  {\begin{enumerate}% Regular enumerate
     \let\olditem\item% Store \item
     \renewcommand{\theenumi}{\thepairitem\ifodd\value{enumi}a\else b\fi}% \item counter + reference
     \renewcommand{\labelenumi}{(\theenumi)}% Counter representation in list
     \setcounter{pairitem}{0}% Restart numbering
     \renewcommand{\item}{% Update \item to check whether enumi is odd (for 'a'/'b' suffic
       \ifodd\value{enumi}\else\stepcounter{pairitem}\fi
       \olditem
     }}
  {\end{enumerate}}

\begin{document}

\begin{pairenumerate}
  \item item 1a
  \item item 1b

  \item\label{foo} item 2a
  \item item 2b
\end{pairenumerate}

Item~\ref{foo} is nice.

\end{document}