Thank you for testing.

I pushed the change to the stable branch (with minor changes to the comment):
https://hg.savannah.gnu.org/hgweb/octave/rev/6fa106924c19

It should be part of Octave 8.4.0 (if that is released) or Octave 9.

Closing as fixed.

Thank you for bringing this to our attention.
Does the following change avoid the issue?

diff -r d313e8ece0b8 src/mkoctfile.in.cc
--- a/src/mkoctfile.in.cc        Sat May 13 13:57:14 2023 +0200
+++ b/src/mkoctfile.in.cc        Sat Aug 26 13:06:17 2023 +0200
@@ -943,9 +943,12 @@
         {
           // Pass through any unrecognized options.
           pass_on_options += (' ' + arg);
-          // Check for an additional argument following the option.
+          // If the current argument does not contain a `=` character, check
+          // for an additional argument following the option.
           // However, don't check the final position which is typically a file
-          if (i < argc-2)
+          // FIXME: We might be skipping to parse some option with this.
+          //        See bugs #52928 and #64590.
+          if (i < argc-2 && (arg.find ('=') == std::string::npos))
             {
               arg = argv[i+1];
               if (arg[0] != '-')

I came across an issue with mkoctfile, which can be exercised with the Makefile attached to this message. When copying it to an empty directory and running "make", it produces the output below.

touch b.cc
touch a.cc
mkoctfile --verbose -c a.cc
g++ -c -Wdate-time -D_FORTIFY_SOURCE=2 -fPIC -I/usr/include/octave-8.3.0/octave/.. -I/usr/include/octave-8.3.0/octave  -pthread -fopenmp -g -O2 -ffile-prefix-map=/build/reproducible-path/octave-8.3.0=. -flto=aut
o -ffat-lto-objects -fstack-protector-strong -Wformat -Werror=format-security    a.cc -o a.o
===== FINE
mkoctfile --verbose b.cc a.o
g++ -c -Wdate-time -D_FORTIFY_SOURCE=2 -fPIC -I/usr/include/octave-8.3.0/octave/.. -I/usr/include/octave-8.3.0/octave  -pthread -fopenmp -g -O2 -ffile-prefix-map=/build/reproducible-path/octave-8.3.0=. -flto=aut
o -ffat-lto-objects -fstack-protector-strong -Wformat -Werror=format-security    b.cc -o /var/tmp/oct-8sYg58.o
g++ -I/usr/include/octave-8.3.0/octave/.. -I/usr/include/octave-8.3.0/octave  -pthread -fopenmp -g -O2 -ffile-prefix-map=/build/reproducible-path/octave-8.3.0=. -flto=auto -ffat-lto-objects -fstack-protector-str
ong -Wformat -Werror=format-security  -o b.oct  a.o /var/tmp/oct-8sYg58.o   -shared -Wl,-Bsymbolic  -L/usr/lib/x86_64-linux-gnu -shared -Wl,-Bsymbolic  -L/usr/lib/x86_64-linux-gnu -shared -Wl,-Bsymbolic  -flto=a
uto -ffat-lto-objects -Wl,-z,relro
===== FAIL
mkoctfile --verbose -fdebug-prefix-map=.=. b.cc a.o
g++ -I/usr/include/octave-8.3.0/octave/.. -I/usr/include/octave-8.3.0/octave  -pthread -fopenmp -g -O2 -ffile-prefix-map=/build/reproducible-path/octave-8.3.0=. -flto=auto -ffat-lto-objects -fstack-protector-str
ong -Wformat -Werror=format-security  -fdebug-prefix-map=.=. b.cc -o a.oct  a.o   -shared -Wl,-Bsymbolic  -L/usr/lib/x86_64-linux-gnu -shared -Wl,-Bsymbolic  -L/usr/lib/x86_64-linux-gnu -shared -Wl,-Bsymbolic  -
flto=auto -ffat-lto-objects -Wl,-z,relro
rm a.cc

This Makefile has two targets "fine" and "fail", whose difference is only the presence (or absence) of option -fdebug-prefix-map when invoking mkoctfile. In both cases, the output file produced by mkoctfile should be b.oct. This is indeed the case for the "fine" target. However, for the "fail" target, file a.oct is built.