Sen. Mike Lee CONFIRMS Vance could take over Thune's spot as Presiding Officer in the Sena

Is this power limited to the majority party? In other words, could a Republican VP assume power even over a Democratic majority Senate?

Interesting note from the Heritage Foundation:

The Constitution seems to suggest that the Vice President is required to preside over the Senate, the text providing that the Vice President “shall be president of the Senate.” But for decades, officeholders have presided very little. In such cases, the Constitution empowers the Senate, in the Vice President’s absence, to select a “President pro tempore.”

The VP might actually have the powers discussed, but traditionally has not exercised it. Also in the article:

The Vice President can make parliamentary rulings, which can be overruled by the Senate.25 For instance, Vice President John Nance Garner made a series of pronouncements to the chamber in the 1930s. He formalized the custom whereby the Majority Leader would be recognized before all other Senators seeking to address the body. These statements, which went unchallenged by the Senate at the time, today undergird the Majority Leader’s principal formal power, which is to be recognized first and to propose the Senate schedule.26

As a practical matter, the Parliamentarian has narrowed the presiding officer’s authority over Senate procedure. This nonpartisan position was created in 1935 and provides technical advice on rules and precedents that the Vice President rarely ignores.