No, tailscale is a VPN in the original sense, your devices act like they're in the same network much like your WiFi printer does at home, regardless of open ports. So it works with any ports and applications, not just ssh.

You might need to change the hostname you are trying to access. If e.g. homeserver.example.com points to your public IP, it still won't work any better. But if your devices is called "homeserver" in tailscale and on your other (client) device you point to http://homeserver/, then the tailscale DNS would resolve it to the tailscale-internal IP and it will route through tailscale.