Have Vice Presidents of the United States typically had more influence when the President they were serving under were Vice Presidents as well?

This is a nice thought, but I think this would very much depend on personality of the leader in question. Lyndon Johnson, for example, had an incredibly forceful and intimidating styles of leadership, and his Vice President Hubert Humphrey was hand-picked as a yes-man and supporter. As I found on Wikipedia and cross-referenced, he would later be quoted in his biography as saying, "After four years as Vice-President ... I had lost some of my personal identity and personal forcefulness. ... I ought not to have let a man who was going to be a former President dictate my future."

Gerald Ford, on the other hand, had a notably lighter touch. When approaching Nelson Rockefeller as a running-mate, he promised to make him a "full partner" during his presidency. Whether he followed through on this promise or not is debatable.

The reason for such disparity is because the "role of the Vice President" is actually incredibly vague in the Constitution: "The Vice President of the United States shall be President of the Senate, but shall have no Vote, unless they be equally divided." (Article I, sec. 3). This, and some later sections about how the Vice President is elected alongside the President, is the only mention of the Vice President in entirety. Strictly speaking, then, the President does not have to delegate any authority whatsoever to his or her Vice President. On the other hand, they are uniquely involved in both the executive and legislative branches, and so are therefore often called upon to head up Senate committees as the President's proxy.