Please help us help you and fill where relevant:
Your LimeSurvey version: 6.5.11
Own server or LimeSurvey hosting: Own Server
Survey theme/template: fruity_twentythree
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We have defined some extra attributes in CPDB as dropdown (e.g. "size_of_company" for better categorization of the results)
The attributes of participants of the survey are assigned.
In registration is only C.Html.textField and it don't seems to be possible to access to the defined dropdown-list?
 

Yes,

There are no included solution for this.
You can update the form via SUrvey theme : github.com/LimeSurvey/LimeSurvey/blob/ma...n/register_form.twig