Here’s why you’re being led astray: you are identifying yourself as a ‘person’ instead of consciousness itself.
Your reasoning appears to be something like the following: “Since I exist as a human person and I am conscious, then the existence of “other people” must entail the existence of others who are like me – human persons who are conscious. And this would make solipsism false.”
The mistake here is the assumption that you exist as a human person. You are not really a human at all -- you are consciousness itself. You exist as the entire perceptual world, and the human person you commonly identify as yourself is nothing but a portion of this perceived world. Chairs, tables, rocks, skies, birds, trees, other people, your computer screen, etc. are also perceived and therefore equally part of ‘yourself’ as this human person.
Think of it like a dream. Most dreams feel as if they’re inhabited by an embodied human person. But the human person is not having the dream – it is part of the dream (and so are the bodies of your dream characters). This is why the existence of ‘other people’ is not at odds with solipsism. As long as it is understood that these people are not metaphysically distinct of yourself (where ‘yourself’ refers to the entirety of conscious experience as opposed to any human person appearing within conscious experience) then other people can be said to exist in the same way dream characters exist in dreams.
Think of solipsism as a continuous dream that is impossible to wake up from.
Because if solipsism is the case then no human person ‘has’ a mind. Human persons would appear within the mind, in the same way (if naturalism is the case) human persons would appear within a naturalistic universe. The mind has human persons; not the other way around.
On solipsism, human beings are p-zombies across the board. Even the human being you commonly identify as yourself – the one appearing at the center of the visual field – would be just as unconscious as any other inanimate object.