There is actually some biological legitimacy to this trope. Aging from birth until the end of puberty is controlled by the release of hormones, such as somatotropin, from the endocrine system, lasting somewhere between 21 and 23 years for most people. Post-puberty, aging is the result of gradual tissue degradation. This is because DNA replication is not perfect, skipping over large portions of genetic information at the end of each chromosome every time a cell divides. In response, our chromosomes have structures called telomeres on their ends to protect the most important components from being cut out.
Telomeres are basically just repeated sections of DNA that can be lost with no consequences. Every time a cell divides, the telomeres in all 46 of its chromosomes get shorter and shorter until they disappear, which is when the cell stops dividing altogether and eventually dies. As more and more of the body's cells lose their telomeres, age-related characteristics become more noticeable. The skin loses elasticity, bone and muscle mass decrease, hair pigment cells decline in number, etc. With immortality, telomeres could be replaced at the same rate they're lost or even faster, preventing the physical effects of aging from ever appearing at all. It's already possible in some species, such as flatworms, who can regenerate their telomeres indefinitely.
Because telomere regeneration has nothing to do with hormone-driven maturation, it is entirely plausible that a biologically immortal human would develop at the same rate as a regular one and plateau once physical peak is reached - that being somewhere around the mid-20s or early 30s.
