Any format released after the specified date is accepted. An earlier release date issues a warning.
Consider the file
\NeedsTeXFormat{LaTeX2e}[2023/11/01] \showthe\fmtversion \stop
No problem at all and I get
> \fmtversion=macro: ->2023-11-01.
Let's fake being after the next release, which will be dated 2024-05-31 (or thereabout):
\def\fmtversion{2024/05/31} \NeedsTeXFormat{LaTeX2e}[2023/11/01] \show\fmtversion \stop
No problem at all and I get
> \fmtversion=macro: ->2024/05/31.
Note that \NeedsTeXFormat will look at the meaning of \fmtversion and nothing else. Indeed if I fake having an earlier release
\def\fmtversion{2023/05/31} \NeedsTeXFormat{LaTeX2e}[2023/11/01] \show\fmtversion \stop
the LaTeX run issues
LaTeX Warning: You have requested release `2023/11/01' of LaTeX, but only release `2023/05/31' is available. > \fmtversion=macro: ->2023/05/31.
latex-base.