Adaptation is a functional notion, not a logical or semantic a priori definition, despite what Popper thought. The current understanding of fitness is dispositional . That is to say, fitness is a disposition of a trait to reproduce better than competitors. It is not deterministic. If two twins are identical genetically, and therefore are equally fit, there is no guarantee that they will both survive to have equal numbers of offspring.
Fitness is a statistical property.
What 'owns' the fitness isn't the organism, but the genes. They will tend to be more often transmitted so far as what they deliver is better 'engineered' to the needs of the organisms in the environment in which they live. Moreover, fitness exists over and above the properties of the individual organisms themselves. There are three debated ways to construe this. Fitness can be a relation of genes to other genes.