Let's see what happens with the first example:
\begin{center}
{\LARGE
Text with\\linebreak}
\end{center}
You open a group (\begin) and order TeX to center lines; in this environment \\ is just \par with some code added to make it obey the usual syntax with *-variant and optional spacing argument.
Then you open another group with { (with \begingroup it would be the same, as long as the group is ended with \endgroup instead of }) in which you issue the \LARGE command.
The fact that \\ does \par means that two paragraphs are typeset; the first is set with the baselineskip pertaining to \LARGE. But the second (with the word linebreak not: indeed the center environment issues a \par to finish off the environment, but this \par comes along when the group started with { has ended and the font size command has been undone. So the second paragraph is typeset with the baselineskip that is in force when the \end{center} is seen, that is, the value it had when the center environment was started. The lines don't overlap because TeX doesn't normally do it (1pt of vertical space will be between the two lines, more precisely the value of \lineskip).
How to solve the issue? Being simple!
\begin{center}
\LARGE
Text with\\
linebreak
\end{center}
The inner group does nothing good and something bad. The group opened by \begin is closed by \end, so the setting of \LARGE will be undone anyway.
If you do it with your env environment, you just add a grouping, but
\begin{env}
Text with\\
linebreak
\end{env}
will work correctly, because it essentially resolves into
\begingroup
\begin{center}
\LARGE
Text with\\
linebreak
\end{center}
\endgroup
(the outer group is provided by \begin{env} and \end{env}), so what I said above applies.
Adding a minipage is probably not what you want: just look at the output of this example to see why:
\documentclass[twocolumn]{article}
\usepackage{lipsum}
\begin{document}
\lipsum*[2]
\begin{center}
\LARGE
Text with\\
linebreak
\end{center}
\lipsum[2]
\newpage
\lipsum*[2]
\begin{center}
\begin{minipage}{\columnwidth}
\centering
\LARGE
Text with\\
linebreak
\end{minipage}
\end{center}
\lipsum[2]
\end{document}

Why is the spacing different? The minipage in the right column has large height and depth, so above and below the minipage TeX uses the \lineskip glue, while in the left column the stated values of \baselineskip are used.
\par? Sorry, no. – egreg Sep 25 '13 at 09:17\parwhen working in grouped scope suffices. My assumption from the OP's post was a very particular case and not necessarily localized to an environment that contains only 2 lines of code in a different font. Your approach is merely to suggesting removing the grouping inside the environment. – Werner Sep 25 '13 at 17:17\\, as you're incenter. – egreg Sep 25 '13 at 17:22