I never found the claim that vampires are entirely different people from the original to be credible. They’re certainly changed - they’re given a new set of goals (feed/kill/hurt), they lose a lot of their capacity for empathy, and so on. But the scale of these changes doesn’t seem like anything far outside of normal human experience. Puberty, for example, brings on a whole new set of behavioral imperatives - I’ve certainly changed since elementary school, but I’d be loathe to say that elementary-school Mr. Excellent was killed and replaced by me. And medical history is replete with folks who’ve suffered brain damage that impaired that capacity for impulse control, caused inappropriate sexual behavior (even a propensity for sexual assualt), and so on. Again, these people are certainly changed - but they’ve not been killed and replaced by dopplegangers.
Further, it’s clear in the Whedonverse that cognition takes place in the brain, whether one is a human or vampire - the Initiative neutered Spike by putting an implant in his brain, and if his brain had been damaged during a botched effort at extraction, he would have risked becoming a vegetable. When Angel was trapped underwater for a summer, Wesley feared that starvation would cause brain damage - which would turn him into a vegetable.
Finally - the vampires themselves seem to believe that there is continuity of consciousness between the original human and the vampire. Angelus tortured Dru before turning her, specifically so that she would remain crazy after she was turned.
Vampires in the Whedonverse are people who’ve been drastically changed - given powerful and dangerous impulses, and stripped of most of their capacity to control them. But they’ve clearly retained continuity of consciousness with their human selves - they’re the same people - changed, but with (as we’ve seen) some residual capacity to control their changed behavior.
Which raises an interesting ethical problem: One of Buffy’s primary modes of operation is to wander graveyards, staking newly raised vampires. In so doing, she’s often killing people who have never done anyone any harm in their lives, nor tried to do so. (Consider the cases in which she’s “helped” a vamp out of his coffin, only to immediately stake him). One could argue that many or most of these vampires will inevitably go on to kill people - but that clearly isn’t so: When faced with situations in which they know that transgression will lead to their death, both Spike and Harmony were capable of controlling their behavior. In fact, once Angel took over Wolfram and Hart, it seems like soul-less vampires were routinely employed on that basis. Vampires may not have much of an intrinsic moral impulse, but they can be deterred, and they then lead fairly normal and harmless lives.
In short, there’s a fine argument to be made that Buffy is a particularly appalling sort of monster.