Hi Dariof,
Indesign interactive form fields do not work in Adobe Sign. It will automatically detect the form fields which will work when you upload the document to Adobe Sign.
If you need those data fields, you can create the form fields in the Adobe Sign.
I have checked the form that you have shared in the link above.
I see that you have assigned all the signature fields to the single signer. You need to assign each signature field to the individual signer.
Take help of the steps provided in this help document Send agreements and collect signatures
When you send the document for signature, please make sure that you check the box for "Preview & Add Signature Fields". Check the screenshot below:

To add and assign the form fields, please refer to the steps mentioned in the following help documents
- Field types
- Assign form fields to recipients
Hope that will help.
Let us know if you have any questions.
Regards,
Meenakshi