How to set up the ssh-agent on sway?

From my understanding, x11 shares stuff among terminals, so at boot I can run the ssh-agent.

However, now I have migrated to wayland (sway) and I am having trouble getting the same behaviour from the ssh-agent.

The expected behaviour:
After booting manjaro, I can run ssh-add and it will prompt me for the passphrase. After entering it, I should be able to use this ssh-key until I restart the machine. Also I should be able to use it in newly spawned terminals. The agent should be shared.

What I have tried so far:
I have tried all kinds of different snippets in my .zshrc. (I was using bash on i3 before, so maybe I am misunderstanding something here itself):

From my understanding, this line is supposed to run the ssh-agent:

eval `ssh-agent -s`

When I have this line, and I use ssh-add, I receive the prompt and the password will be stored but it won’t be shared with other terminals.

When checking the arch wiki I came across this snippet:

if ! pgrep -u "$USER" ssh-agent > /dev/null; then
    ssh-agent -t 1h > "$XDG_RUNTIME_DIR/ssh-agent.env"
fi
if [[ ! -f "$SSH_AUTH_SOCK" ]]; then
    source "$XDG_RUNTIME_DIR/ssh-agent.env" >/dev/null
fi

While browsing the web, I have come across similar snippets like these:

if ! pgrep -u "$USER" ssh-agent > /dev/null; then
    ssh-agent -t 1h > "$XDG_RUNTIME_DIR/ssh-agent.env"
fi

or this one:

if [ -z "$SSH_AUTH_SOCK" ] ; then
 eval `ssh-agent -s`
 ssh-add ~/.ssh/id_rsa
fi

But none seem to work for me.

Also when I use echo $SSH_AUTH_SOCK I noticed that the output is different in each terminal that I have opened. Shouldn’t this point to the same sock?

Has anyone managed to produce the behaviour I described? Because I used to have it like that when using i3.