Do I need to explicitly map my website domain in Pushwoosh or Firebase when setting up web push notifications?

No, there is no separate "domain mapping" step. Your website is bound to Pushwoosh through the web push platform configuration plus the Web SDK served from your site.

How web push is configured

• Go to Settings → Platform Configuration → Web Push Notifications and click Configure. One configuration covers all modern browsers, including current Safari versions; only legacy Safari (macOS 10.14–12) needs a separate setup.
• Web push uses VAPID keys, not Firebase server credentials. Either choose Create a new push configuration and let Pushwoosh generate a VAPID key pair, or choose Use existing configuration and paste your keys into Web Push Certificate (public key) and Web Push Private Key — useful when migrating from another service so your existing subscribers do not have to opt in again.
• If you are reusing Firebase keys, take them from Firebase Console → Project settings → Cloud Messaging → Web Configuration (use the three-dot menu → Show private key for the private half).
• The legacy FCM Server Key / Sender ID pairing is no longer used for web push — Google discontinued legacy FCM server keys in June 2024.

What actually ties the configuration to your domain

• The browser binds the subscription to the origin the service worker was registered from, so the Pushwoosh service worker file and the page loading the SDK must be served from the same HTTPS origin.
• Subscriptions do not carry over between different origins (including between a domain and its subdomains), so make sure you integrate the SDK on the exact host you want to collect subscribers on.

For details, see the Web push configuration guide and the Web Push SDK integration guide.