Hi Boudewijn,
Does the "customer portal" of your customer refer to microsoftcrmportal or powerappsportal? or is it website hosting on his own server?
No matter which approach it takes, however, the only supported way to trigger "Marketing List subscribed/unsubscribed" behavior is by using embedded Subscription Center Marketing Form, in other word, only Marketing Form generated by Marketing application has capability to log contacts' interaction.
If the "customer portal" is Portal service provided by Microsoft: please connect your Marketing application to Event Portal, because only the event portal supports marketing functionalities.
If it refers to an external webpage, then you can use embedded Subscription Center Marketing Form to build your customer subscription center.
Steps to set up an external Subscription Center: (You also could take documentation process as reference.)
1. Authenticate your domain.
Domain authentication (Dynamics 365 Marketing) | Microsoft Docs
2. Create a subscription center type marketing form, after going live it, create a new form page record in Form hosting tab.
3. In the new form page record, copy the automatically generated script.
4. Paste the script to page of the "customer portal".
5. Create a new content settings record, set Subscription center field to the URL of "customer portal" page.
6. Before going live a new customer journey, set Content settings field(in General tab) to the new custom content settings.
Then contacts will be navigated to your own subscription center page after they clicking the subscription link in email, then they can manage their subscription(marketing) list.
Regards,
Clofly
