A fetus is for one thing alive. Second if you use the definition of being a person as "a mammal belonging to the species Homo Sapiens then yes because the fetus has Homo Sapiens DNA. So a fetus is both alive and a person. By the way legal biological statuses really should be replaced with medical statuses; it would make things much more logical.
Fetuses are people but not at first. It takes time for them to develop into a person. Once the fetus has a heartbeat and brain it should be considered a person. Before that I do not think it is a person This brings up a large topic of when is abortion okay.
It is scientifically proven that a foetus does not develop nerve endings until the 28th week in the womb. When we are born, we have no recollection or memory of our time in the womb because our brains are not fully developed. A foetus cannot survive outside of the womb up to a minimum of 24 weeks, and most babies born at 24 weeks die. They cannot survive outside of the womb because they don't haven't developed organs or are sometimes missing them altogether. A human feels, thinks with a re-collective memory and can survive because they have fully developed systems that allow them to survive.
A fetus before the viability point has no consciousness (nor the brain development necessary to be conscious), no sentience (again, nor the brain development to be sentient), and cannot feel pain (does not have the neurological development for pain reception).
Without these characteristics, there are none of the hallmarks of personhood and no good reason to make them "people" under the law.
In my opinion fetuses are not people and it should never be implied that they are people. They are fetuses and as such are in the growth stages toward becoming people, but are not developed people. That is why we have birth records and certificates. Not conception records and fertilization certificates.