The affiliate sign-up page can be customized in the following ways:
The color of the form (works based on the branding color you selected)
The logo on the form (uses the same logo as you added to the platform)
The language of the registration (works based on the system language you selected in Settings > General > System > Language section)
Text on the right side (this text can be updated by navigating to Settings > Affiliates > Affiliate communication)
Text when affiliates are pending and are on the awaiting approval page (this text can be updated by navigating to Settings > Affiliates > General)
Extension of registration with custom registration data (this can be set up by navigating to Settings > Affiliates > Affiliate registration data)
Alternatively, there is a way to create affiliates via API, thus you could have a fully branded custom page for the registration from your side and then redirect user to the platform after registration.
One thing to note for the API creation of affiliates - it currently does not allow adding of custom registration data (the last bullet point mentioned above).
This is the endpoint needed: https://developers.trackdesk.com/#post-/api/node/affiliates/v1/register-with-user