Different output with normal environment & newenvironment when citing

There are a few misunderstanding of how environments work.

First off, notwithstanding that newenvironment{foo}{...}{...} defines foo and endfoo doesn't mean that one can use foo and endfoo in the document. To the contrary, they should not.

Second problem. The command newenvironment requires three mandatory arguments: the name of the environment, the code to be executed at the start and the code to be executed at the end. The parameter #1 in the second mandatory argument does not refer to the environment contents.

Your code

newenvironment{qt}[1]
      {begin{quote}
      {enquote{{#1}}}
      end{quote}
      }

misses the final mandatory argument and, since it's followed by a blank line, the “end part” is taken to be par. Let's see how the call

qt
{Lorem ipsum dolor enquote{sic} amet.}
cite{key}
endqt

is processed. The macro qt is defined to have an argument, which is found to be the braced group following. Then the “begin part” is inserted and we find

begin{quote}{enquote{{Lorem ipsum dolor enquote{sic} amet.}}}end{quote}
cite{key}
endqt

Now the code is processed normally, finding cite{key} outside quote. Finally par results from endqt.

How to fix?

documentclass{article}
usepackage[romanian]{babel}
usepackage[backend=biber]{biblatex}
usepackage{csquotes}
DeclareQuoteStyle{romanian}
  {quotedblbase}
  {textquotedblright}
  {guillemotleft}
  {guillemotright}

newenvironment{qt}[1]
 {begin{quote}enquote{#1} ignorespaces}
 {end{quote}}

begin{filecontents}{jobname.bib}
@book{key,
 title        = {Title},
 author       = {Author, An},
 year         = {2021},
 location     = {Location},
 publisher    = {Publisher}
}
end{filecontents}
addbibresource{jobname.bib}

begin{document}

With textbackslash newenvironment:
begin{qt}{Lorem ipsum dolor enquote{sic} amet.}
cite{key}
end{qt}

With normal environment:

begin{quote}
enquote{Lorem ipsum dolor enquote{sic} amet.}
cite{key}
end{quote}

printbibliography
end{document}