History absolutely is recorded subjectively. What is more, there is no other way that history could be recorded.
To be sure, the factual details of history are generally recorded objectively. There is no dispute as to, for example, the date on which the Japanese attacked Pearl Harbor in 1941. However, the important part of history is in the "whys" of what happened and in the impacts of those events. This is where history cannot be recorded objectively.
For example, a major question in history has to do with why the United States used the atomic bombs on Japan. This is something that can never be objectively known. Different US policy makers may have had different reasons. Reasons stated publicly might not be the real reasons. Historians have to sift through the evidence and decide for themselves. There is no way to do this objectively, without taking into account a historian's own biases or opinions.
Since the parts of history that matter are largely matters of opinion, there is no way that history can be written objectively. Therefore, it is always recorded in a more or less subjective way.