Where is terminal output stored

This is a simple solution if you're willing to accept output just above the current prompt line.

bufferout () {
    local buffer
    while IFS= read -r line; do              # buffer stdin
        buffer="$buffer$line\n"
    done
    print -rn -- $terminfo[dl1]              # delete current line
    printf "$buffer"                         # print buffer
    kill -USR1 $$                            # send USR1 when done
}

TRAPUSR1 () {                                # USR1 signal handler
    zle -I                                   # invalidate prompt
    unhash -f TRAPUSR1 bufferout             # remove ourselves
}

./testout 2>&1 | bufferout &!                # run in background, disowned

When the job is done, the current prompt and input buffer will be removed and the entirety of the command's stdout and stderr will be printed.

Note that this assumes it will be run exactly once and removes itself afterwards. If you want to keep reusing this functionality in the same shell, remove the unhash -f line in TRAPUSR1.

This answer includes an improvement suggested by Clint Priest in the comments. Thanks!

A linux kernel should store an on-screen log for your vts in the corresponding /dev/vcsa*[ttynum] device.

It is why the following works:

echo hey >/dev/tty2
dd bs=10 count=1 </dev/vcs2

...which prints...

hey       

The corresponding /dev/vcsa[ttynum] device will store an encoded version of the formatted text on-screen, whereas the /dev/vcs[ttynum] will be a plain dump. The vcsa[ttynum] devices will encode a pair of bytes which describe each on-screen char and its attributes, as well as a string at the head of each logical page that indicates the referenced tty's lines,columns count.

As @kasperd points out, I had it wrong before by assuming the \a BEL was encoded between every character, when in fact: The default color combination happens to coincide with the bell character.

For your purposes using the /dev/vcs[ttynum] is probably easiest. Here's a look at the differences:

echo hey >/dev/tty2
dd bs=10 count=1 </dev/vcs2 |
sed -n l

...prints...

hey       $

...and...

echo hey >/dev/tty2
dd bs=10 count=1 </dev/vcsa2 |
sed -n l

...prints...

0\200\000\004h\ae\ay\a$