Tikz picture inside a box

I might have misunderstood the question, but here is a try:

\documentclass{article}

\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}

\usepackage{amsmath}
\usepackage{tikz}
\usetikzlibrary{positioning}


\begin{document}

\begin{figure}
    \centering%
    \begin{tikzpicture}
        \node at (0,0) (a) {a};
        \node at (1,0) (b) {b};
        \draw (a)--(b);

        \node[above = \baselineskip of current bounding box] {%
            \begin{minipage}{0.9\linewidth}
                Text or whatever you want to put here.

                \begin{equation*}
                    1 = 2
                \end{equation*}
            \end{minipage}%
        };
        \draw (current bounding box.north west) rectangle (current bounding box.south east);
    \end{tikzpicture}
\end{figure}

\end{document}

Quick explanations: once the main elements of the picture are drawn, I create a node that gets positioned right on top of everything (above the “current bounding box”, by a distance relative to \baselineskip because it's slightly less dirty). Finally, I draw a rectangle around the final bounding box. … It's a bit overkill, now that I think about it.

By the way, I'm pretty sure tikzpictures can be put in tcolorboxes, since I used tcolorboxes to show examples in the documentation of a pseudo-package I wrote, and these were full of tikz stuff: http://www.alicem.net/files/oths/pointbox.zip

You can include tikzpicture in tcolorbox, at least if you use listings. I know that you said not to use minipage, but not why noy to use it. To me this is the natural start of such a problem, so I included a solution for that as well. And I expanded the MWE with some lipsum text and boxes around nodes.

\documentclass{article}
\usepackage{lipsum}
\usepackage{calc}
\usepackage{tikz}
\usepackage[listings]{tcolorbox}
\newtcblisting{EvalBox}[2][]{%
  colback=white,
  arc=0pt,
  boxrule=0.5pt,
  text only,
  title=#2,#1}
%% 
\begin{document}
\lipsum[1]
%% First
\begin{EvalBox}{}
  \lipsum[2]
  \begin{equation}
    1=2
  \end{equation}

  \centering
  \begin{tikzpicture}
    \node[draw] at (0,0) (a) {a};
    \node[circle,draw] at (1,0) (b) {b};
    \draw[->] (a)--(b);
  \end{tikzpicture}
\end{EvalBox}

%% Second version, using minipage
\noindent
\fbox{\begin{minipage}[t]{1.0\linewidth-2\fboxsep-2\fboxrule}
  \lipsum[2]
  \begin{equation}
    1=2
  \end{equation}

  \centering
  \begin{tikzpicture}
    \node[draw] at (0,0) (a) {a};
    \node[circle,draw] at (1,0) (b) {b};
    \draw[->] (a)--(b);
  \end{tikzpicture}    
\end{minipage}}
\end{document}

EDIT

A version with tcolorbox not using listing can also be done:

\documentclass{article}
\usepackage{lipsum}
\usepackage{tikz}
\usepackage{tcolorbox}
\begin{document}
\lipsum[1]
\begin{tcolorbox}[colback=white]
  \lipsum[2]
  \begin{equation}
    1=2
  \end{equation}
  \centering
  \begin{tikzpicture}
    \node[draw] at (0,0) (a) {a};
    \node[circle,draw] at (1,0) (b) {b};
    \draw[->] (a)--(b);
  \end{tikzpicture}
\end{tcolorbox}
\end{document}

Here is something that doesn't use minipages (for this particular problem).

\documentclass{article}

\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}

\usepackage{amsmath}
\usepackage{tikz}
\usetikzlibrary{positioning, fit}


\begin{document}

\begin{figure}
    \centering%
    \begin{tikzpicture}
        \node [text width = 0.9\linewidth] (first) {
            Text or whatever you want to put here. 
        };
        \node [below=\baselineskip] at (first) (eq) {$1=2$};
        \node [below left=\baselineskip] at (eq) (a) {a};
        \node at ([xshift=1cm] a) (b) {b};
        \draw (a)--(b);
        \node [draw, rectangle, fit=(first)(eq)(a)(b)]{};   
    \end{tikzpicture}
\end{figure}

\end{document}