They don’t identify with a single biological system, even though they may have chosen to focus on one for their studies.
The answer to the question posed in the title is clear: It’s both. When asking this question to new graduate students or some seasoned investigators, it’s not unusual to hear them declare that biochemistry is “just a tool.” Indeed, biochemists have developed methods and reagents that are essential for other disciplines.
Strictly speaking, a biochemist is someone who studies the underlying chemistry of biological processes and systems.
Traditionally, biochemistry encompasses the study of the chemistry essential to biological processes including, but not restricted to, enzymes, metabolism and signal transduction.