Actual Content of a Symlink File in ext4

With the symlinks utility by Mark Lord (offered by many distributions; if yours doesn't have it, build it from source):

chroot /home/user/system symlinks -cr .

Alternatively, on systems that have a readlink command and a -lname predicate to find (warning: untested code):

cd /home/user/system &&
find . -lname '/*' -exec ksh -c '
  for link; do
    target=$(readlink "$link")
    link=${link#./}
    root=${link//+([!\/])/..}; root=${root#/}; root=${root%..}
    rm "$link"
    ln -s "$root${target#/}" "$link"
  done
' _ {} +

now bin-xyz is a broken link

The reason for that is hopefully obvious: Either .. has to refer to the actual parent directory or else it would be completely ambiguous.

Since ~/a/src -> /home/user/b/sync/src, I presume with that as $PWD and ln -s ../bin bin-xyz you're trying to link /home/user/b/sync/src/bin-xyz to /home/user/a/sync/bin. In which case you should just use absolute paths:

 ln -s /home/user/a/sync/bin /home/user/b/sync/src/bin-xyz

The reason ln doesn't fill in the .. blanks in this regard is that it, unlike your current shell, has not been tracking your movement and has no way to know that by .. you meant something other than the real parent directory.

It's (maybe?) implied in your question that you can't do this because there are actually two different systems involved. That implies either a network share is mounted or you use the same block device with both, implying in turn an XY problem -- can't solve that without more details.