After requesting the link, when using Mail or Gmail via Safari, after you long press on the sign-in link, you should be prompt with the following screen
You should see the "open in PhotoRoom" option. If you tap on it, you should be redirected to the app.
If the link is too old, then you might be prompted with an error message in PhotoRoom app. Don't worry, in that case, just resend a link to your email address and you should be fine.