Relevant bit of the source:
...
    36    try
    37      {
    38        throw_exception_1 (10);
    39      }
    40    catch (const int *e)
    41      {
    42          std::cerr << "Exception #" << *e << std::endl;
    43      }
    44    i += 1; /* Break after exception 1.  */
...

The -m64 scenario in more detail:
- the test-case runs to throw_exception_1.
- it installs a FinishBreakpoint, which is a temporary breakpoint set at the
  return address of a frame.
- for -m64, that address is:
    400c47:       83 45 e4 01             addl   $0x1,-0x1c(%rbp)
  which corresponds the "i += 1 at line 44"
- the test-case then continues
- an exception is throw in throw_execution_1
- the exception is caught at line 40, and a message is printed
- line 44 is executed, and the FinishBreakpoint triggers.

With -m32, we have instead:
- the address where the finish breakpoint is set is:
    8048a0a:       83 c4 10                add    $0x10,%esp
  which is the lasn insn generated for the call at line 38
- the test-case continues
- an exception is throw in throw_execution_1
- consequently, the FinishBreakpoint is not triggered.

In conclusion, the test works by accident for -m64, because the first insn
after the call to throw_exception_1 is also the first insn after the try.
And that just happens to be not the case for -m32.

[ This was also noted by Andrew here ( https://sourceware.org/pipermail/gdb-patches/2012-September/096347.html ):
...
A further issue is that the testing for FinishBreakpoints, in gdb/testsuite/gdb.python/py-finish-breakpoint2.exp, the test action titled "check FinishBreakpoint in catch()" expects the "stop" method to fire rather than the "out_of_scope" method, this is due to the generated code (on x86 and maybe other targets), the first breakpoint we hit after throwing the exception happens to be the finish breakpoint, however this is not guaranteed, and means that (a) the test does not match the documentation, and (b) the test is platform specific.
... ]

Created attachment 14521 [details]
Tentative patch

(In reply to Tom de Vries from comment #2)
> Created attachment 14521 [details]
> Tentative patch

Hmm, reading over Andrew's submission, I've come to the same solution as him, the tentative patch just tries to solve less problems.

https://sourceware.org/pipermail/gdb-patches/2022-December/194821.html

The master branch has been updated by Tom de Vries <vries@sourceware.org>:

https://sourceware.org/git/gitweb.cgi?p=binutils-gdb.git;h=64760036a846099158bb2bab5370ae033dde8db0

commit 64760036a846099158bb2bab5370ae033dde8db0
Author: Tom de Vries <tdevries@suse.de>
Date:   Sat Dec 31 08:51:40 2022 +0100

    [gdb/python] Fix gdb.python/py-finish-breakpoint2.exp for -m32
    
    [ Partial resubmission of an earlier submission by Andrew (
    https://sourceware.org/pipermail/gdb-patches/2012-September/096347.html ), so
    listing him as co-author. ]
    
    With x86_64-linux and target board unix/-m32, we have:
    ...
    (gdb) continue^M
    Continuing.^M
    Exception #10^M
    ^M
    Breakpoint 3, throw_exception_1 (e=10) at py-finish-breakpoint2.cc:23^M
    23        throw new int (e);^M
    (gdb) FAIL: gdb.python/py-finish-breakpoint2.exp: \
      check FinishBreakpoint in catch()
    ...
    
    The following scenario happens:
    - set breakpoint in throw_exception_1, a function that throws an exception
    - continue
    - hit breakpoint, with call stack main.c:38 -> throw_exception_1
    - set a finish breakpoint
    - continue
    - hit the breakpoint again, with call stack main.c:48 -> throw_exception
      -> throw_exception_1
    
    Due to the exception, the function call did not properly terminate, and the
    finish breakpoint didn't trigger.  This is expected behaviour.
    
    However, the intention is that gdb detects this situation at the next stop
    and calls the out_of_scope callback, which would result here in this test-case
    in a rather confusing "exception did not finish" message.  So the problem is
    that this message doesn't show up, in other words, the out_of_scope callback
    is not called.
    
    [ Note that the fact that the situation is detected only at the next stop
    (wherever that happens to be) could be improved upon, and the earlier
    submission did that by setting a longjmp breakpoint.  But I'm considering this
    problem out-of-scope for this patch. ]
    
    Note that the message does show up later, at thread exit:
    ...
    [Inferior 1 (process 20046) exited with code 0236]^M
    exception did not finish ...^M
    ...
    
    The decision on whether to call the out_of_scope call back is taken in
    bpfinishpy_detect_out_scope_cb, and the interesting bit is here:
    ...
                 if (b->pspace == current_inferior ()->pspace
                     && (!target_has_registers ()
                         || frame_find_by_id (b->frame_id) == NULL))
                   bpfinishpy_out_of_scope (finish_bp);
    ...
    
    In the case of the thread exit, the callback triggers because
    target_has_registers () == 0.
    
    So why doesn't the callback trigger in the case of the breakpoint?
    
    Well, the b->frame_id is the frame_id of the frame of main (the frame
    in which the finish breakpoint is supposed to trigger), so AFAIU
    frame_find_by_id (b->frame_id) == NULL will only be true once we've
    left main, at which point I guess we don't stop till thread exit.
    
    Fix this by saving the frame in which the finish breakpoint was created, and
    using frame_find_by_id () == NULL on that frame instead, such that we have:
    ...
    (gdb) continue^M
    Continuing.^M
    Exception #10^M
    ^M
    Breakpoint 3, throw_exception_1 (e=10) at py-finish-breakpoint2.cc:23^M
    23        throw new int (e);^M
    exception did not finish ...^M
    (gdb) FAIL: gdb.python/py-finish-breakpoint2.exp: \
      check FinishBreakpoint in catch()
    ...
    
    Still, the test-case is failing because it's setup to match the behaviour that
    we get on x86_64-linux with target board unix/-m64:
    ...
    (gdb) continue^M
    Continuing.^M
    Exception #10^M
    stopped at ExceptionFinishBreakpoint^M
    (gdb) PASS: gdb.python/py-finish-breakpoint2.exp: \
      check FinishBreakpoint in catch()
    ...
    
    So what happens here?  Again, due to the exception, the function call did not
    properly terminate, but the finish breakpoint still triggers.  This is somewhat
    unexpected.  This happens because it just so happens to be that the frame
    return address at which the breakpoint is set, is also the first instruction
    after the exception has been handled.  This is a know problem, filed as
    PR29909, so KFAIL it, and modify the test-case to expect the out_of_scope
    callback.
    
    Also add a breakpoint after setting the finish breakpoint but before throwing
    the exception, to check that we don't call the out_of_scope callback too early.
    
    Tested on x86_64-linux, with target boards unix/-m32.
    
    Co-Authored-By: Andrew Burgess <aburgess@redhat.com>
    PR python/27247
    Bug: https://sourceware.org/bugzilla/show_bug.cgi?id=27247

Fixed by commit.