Yes, sort of.
From a VSCode terminal run the command
env | grep VSCODE_IPC_HOOK_CLI
then copy-and-paste that line that line with export into your ssh terminal.
After that, you should be able to run code from your ~/.vscode-server/bin/XXX/bin directory.
VSCode terminal

SSH terminal

Update:
You can to automate this with a .bashrc and .profile to place the IPC code into a temp file, and source that when you do your ssh login.
For example, this works for me...
Append this to ~/.bashrc
#
if [ "$VSCODE_IPC_HOOK_CLI" != "" ]; then
cat >$HOME/.vscode_env.sh <<EOF
#
if [ "\$VSCODE_IPC_HOOK_CLI" = "" ]; then
export VSCODE_IPC_HOOK_CLI="$VSCODE_IPC_HOOK_CLI"
alias code="${VSCODE_GIT_ASKPASS_NODE%/*}/bin/code"
fi
EOF
fi
And append this to your ~/.profile
[ -f $HOME/.vscode_env.sh ] && . $HOME/.vscode_env.sh
(There may be more elegant ways. And you still have to start at least 1 terminal in your remote VSCode session.)