We address the identity of biological organisms at play in experimental and modeling practices. We first examine the central tenets of two general conceptions, and we assess their respective strengths and weaknesses. The historical conception, on the one hand, characterizes organisms' identity by looking at their past, and specifically at their genealogical connection with a common ancestor. The relational conception, on the other hand, interprets organisms' identity by referring to a set of distinctive relations between their parts, and between the organism and its environment. While the historical and relational conceptions are understood as opposed and conflicting, we submit that they are also fundamentally complementary. Accordingly, we put forward a hybrid conception, in which historical and relational (and more specifically, organizational) aspects of organisms' identity sustain and justify each other. Moreover, we argue that organisms' identity is not only hybrid but also bounded, insofar as the compliance with specific identity criteria tends to vanish as time passes, especially across generations. We spell out the core conceptual framework of this conception, and we outline an original formal representation. We contend that the hybrid and bounded conception of organisms' identity suits the epistemological needs of biological practices, particularly with regards to the generalization and reproducibility of experimental results, and the integration of mathematical models with experiments.