If you really want larger font size in the display, you can avoid \mathlarger, but I'd not do it: a math display is already prominent enough.
In any case, you need \text for textual parts in math material.
\documentclass{article}
\usepackage{amsmath,amssymb}
\usepackage{etoolbox}
\usepackage{lipsum}
\makeatletter
% see https://tex.stackexchange.com/a/288889/4427
% detach \eqref processing from \tag processing
\let\tagform@ref\tagform@
\let\maketag@@@ref\maketag@@@
\patchcmd{\eqref}{\tagform@}{\tagform@ref}{}{}
\patchcmd{\tagform@ref}{\maketag@@@}{\maketag@@@ref}{}{}
% redefine \tagform@ to use the current size
\def\tagform@#1{%
\maketag@@@{\saved@size(\ignorespaces#1\unskip@@italiccorr)}%
}
% in general \saved@size does nothing
\def\saved@size{}
% now we define a wrapper
\NewDocumentEnvironment{largerdisplay}{O{\large}mb}
{% #1 = desired size (default \large), #2 = type of display, #3 = body
\protected@edef\saved@size{\fontsize{\f@size}{0}\selectfont}%
$$\begin{minipage}{\displaywidth}
\abovedisplayskip=0.8ex \belowdisplayskip=\z@
#1\noindent
\begin{#2}#3\end{#2}%
\end{minipage}$$}{\ignorespacesafterend}
\makeatother
\begin{document}
\lipsum[1][1-4]
\begin{largerdisplay}{align}
& \text{Fie } X(N) = x_0, x_1, x_2, \dots, x_{N-1} \
& \text{unde }N \in \mathbb{N} \text{ și } k \in {0,1,2,\dots,N/2} \
& \text{Avem:}
\end{largerdisplay}
\lipsum[2][1-4]
\end{document}

Compare with the simple align:

\mathlarger– David Carlisle Jun 16 '22 at 22:40şthere as well) but this one got more ↑ now. – user202729 Jun 17 '22 at 07:49