Update after the edit to the question
Two options; in the first one, a vertical strip is used for the title and the vertical strip's length will vary according to the title's length ; in the second option, the vertical strip will be as long as the frame's height.
\documentclass{article}
\usepackage[tikz]{mdframed}
\usepackage{lipsum}
\definecolor{greentitle}{RGB}{165,224,168}
\newenvironment{myenvi}[1]
{\begin{mdframed}[
bottomline=false,
leftline=false,
linecolor=greentitle,
innerrightmargin=25pt,
singleextra={
\node[overlay,anchor=south east,fill=greentitle,rotate=90,font=\color{white}\scshape] at (P) {#1};
},
firstextra={
\node[overlay,anchor=south east,fill=greentitle,rotate=90,font=\color{white}\scshape] at (P) {#1};
},
]
}
{\end{mdframed}}
\newenvironment{myenvii}[1]
{\begin{mdframed}[
bottomline=false,
leftline=false,
linecolor=greentitle,
innerrightmargin=25pt,
singleextra={
\fill[greentitle] (P) rectangle ([xshift=-15pt]P|-O);
\node[overlay,anchor=south east,rotate=90,font=\color{white}\scshape] at (P) {#1};
},
firstextra={
\fill[greentitle] (P) rectangle ([xshift=-15pt]P|-O);
\node[overlay,anchor=south east,rotate=90,font=\color{white}\scshape] at (P) {#1};
},
]
}
{\end{mdframed}}
\begin{document}
\begin{myenvi}{the frame title}
\lipsum[2]
\end{myenvi}
\begin{myenvii}{the frame title}
\lipsum[2]
\end{myenvii}
\end{document}
The result:

Initial version
Like this?

The code:
\documentclass{article}
\usepackage[tikz]{mdframed}
\usepackage{lipsum}
\makeatletter
\protected\def\vvv#1{\leavevmode\bgroup\vbox\bgroup\xvvv#1\relax}
\def\xvvv{\afterassignment\xxvvv\let\tmp= }
\def\xxvvv{%
\ifx\tmp@sptoken\egroup\ \vbox\bgroup\let\next\xvvv
\else\ifx\tmp\relax\egroup\egroup\let\next\relax
\else
%\hbox{\tmp}%original
\hbox to 1.1em{\hfill\tmp\hfill}% centred
\let\next\xvvv\fi\fi
\next}
\makeatother
\newenvironment{myenv}[1]
{\begin{mdframed}[
topline=false,
leftline=false,
innerrightmargin=40pt,
singleextra={
\node[overlay,anchor=north east] at (P) {\vvv{#1}};
},
firstextra={
\node[overlay,anchor=north east] at (P) {\vvv{#1}};
},
]
}
{\end{mdframed}}
\begin{document}
\begin{myenv}{The~frame title}
\lipsum[2]
\end{myenv}
\end{document}
I used David Carlisle's answer to Vertical text (not in table) to write the title vertically.