Should you be able to lay the two side by side and they will talk to each other item for item?
No- I'm not sure that I understand your question.
The risk management plan is a plan - how you will manage risk. It should contain nothing that is ephemeral. Generally it contains sections that describe how you will identify risk, how you will analyze risk, calibration standards (what does a likelihood of 4 mean? What is an impact of 2?). The plan should outline who is involved in managing risks, what roles, what responsibilities. How will risk reporting be done? etc.
The risk registry is a list of risks (usually with some analysis). Everything it contains is ephemeral. Often the registry is a spreadsheet with columns for risk, Likelihood, Impact, mitigation, and owner.
I'm not sure what you mean by talk to each other item for item... the sections of the risk management plan are similar to the sections for all other planning documents. The items in the risk registry are risks.
The risk management plan describes the who, what, where, when, how, and why as it relates to managing threats against your project. It should point to the risk register as one of several tools that would be used to manage risks. The plan would indicate who owns it, who can update it, where it is maintained, how often it might get updated, and how it feeds into whatever reports you plan on publishing.
Your register should be in a constant state of flux while your plan is relatively stable, only updating it as necessary.
Risk management is identifying and evaluating the organization's risk and selecting and implementing measures to treat risk or loss exposures while risk register is a document or tool used to record numbers of risk exposures, regulate compliance and state the relevant information relating to risk such as nature of risk,mitigation and measure and who's responsible for it