the stakeholder register is part of the stakeholder management plan which is an input to the manage stakeholder engagement process.
please check the section "13.2.3.1 Stakeholder Management Plan" it states the below:
The stakeholder management plan is a component of the project management plan (Section 4.2.3.1) and identifies the management strategies required to effectively engage stakeholders. The stakeholder management plan can be formal or informal, highly detailed or broadly framed, based on the needs of the project.
In addition to the data gathered in the stakeholder register, the stakeholder management plan often provides: