Firebase functions + Express: send cookie with response

Question

I'm having a website on firebase hosting and a cloud functions in firebase functions in order for users to login.

I'm calling the /login from the frontend by:

await fetch("https://us-central1-xxxx-website.cloudfunctions.net/api/login", {
                method: "POST",
                credentials: "include", 
                headers: {
                    "Accept": "application/json",
                    "Content-Type": "application/json"
                },
                body: JSON.stringify({ token })
            })

As you can see I include the credentials, and I use the cloudfunctions.net domain. I also tried by using my own domain name, but same result.

In my function I want to send the session token as a cookie:

app.post("/login", async (req, res) => {
...
res.cookie("__session", sessionCookie, {
      expires: new Date(new Date().getTime() + expiresIn), // Add 2 weeks (in milliseconds) to the current epoch
      httpOnly: false,
      secure: true,
      sameSite: 'lax',
      domain: req.get('host')
    });
return res.sendStatus(200);

However on browser side I get:

Same as in the frontend, I also tried using origin in domain, or even explicitly set either my own domain or the cloudfunctions.net domain. In all cases and combination the cookie can't be set. I tried samesite none, same.

I have no 3rd party blocker setup, and this works on another project of mine. I don't get what is wrong there...