Currently, it's not possible to disable the share button in the customer portal, but as a workaround, if your customers are not in an organization, you can select the option "Other customers in their organization. This option only allows them to search for customers in their organization", so if they are not in an organization, they won't find anyone. You can find this configuration on Project settings > Customer permissions.
Related to disable the share button, we have a feature request suggesting the implementation of this ability:
Please, click on vote and watch in order to hear about updates on the development cycle.
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