If it is true that stakeholder register is a part ok Stakeholder Management Plan, then it is contradictory and redundant that "Plan Stakeholder Management" receives the Project Management Plan (including Stakeholder Management Plan) and Stakeholder register.....It seems that Stakeholder register is received twice here....

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:

I think the distinction here is the difference between planning and doing. Managing stakeholders is done during execution with a focus on managing expectations and dealing with concerns and issues. The key stakeholders you will be "managing" has already been captured in the Stakeholder Management Plan.

It is quite unexpected to see that the stakeholder register is NOT an input to the Manage Stakeholder Engagement process. Any reason why this is so? The stakeholder register contains valuable information to facilitate the management of stakeholder relationships.