In her case, she has both rights and privileges under the Constitution. Our Founders recognized that the powerful executive branch they were creating was a risk to the proper authority of Congress. They included specific privileges for members of Congress.
The protection written into the Constitution does not contain an "identification" exception. The burden on the police is total. They may not detain a member of Congress without probable cause that they have committed a serious crime.
Congress itself has complete power to hold their members responsible. They can expel a member, despite the decision of the electorate, if they deem the matter serious enough.
Prior to stopping her, the police had no information indicating that the Congresswoman was guilty of "treason, felony" or "breach of the peace". There is nothing treasonous, felonious, or breaching the peace associated with a representative attending a session of Congress.
For example, it would be entirely unConstitutional for the executive branch to decide what constitutes proper identification of a member of Congress. Otherwise, the executive branch could exclude members that they didn't like simply by denying proper credentials.
What sort of punishment do you think would be appropriate for a police officer stopping a member of Congress from attending its session lacking probable cause that the member had committed treason, felony, or breach of the peace?
If the Supreme Court became involved to decide who, if anyone, should be punished, what do you think they would decide and why?
Consider the case of the protections for the Presidency. Recent precedent has made clear that the President is virtually unaccountable for crimes he might commit, if Congress refused to first remove him from office through the impeachment process. The President's power to pardon for crimes committed contains no exception for the President pardoning himself. The only way to hold him to account while he is in office, is first to remove him from office.
The immunity from arrest for members of Congress is a similar protection. It simply reflects the fact that it is more important to maintain the government in the face of a threat than to hold the duly elected officials to immediate account for misbehavior.
amazing. why don't we all beat up the DC cops for stopping us to identify ourselves at the Capitol building. apparently, it is ok with the grand jury. it might even be fun.