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Is it true that for any two vectors $\vec{AP}$ and $\vec{BP}$ , the angle between $\vec{AP}$ and $\vec{BP}$ will be the same as the angle between $\vec{PA}$ and $\vec{PB}$ ?

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yes, angle between vectors is given as $$\cos\theta=\frac{\vec{AP}\cdot\vec{BP}}{|\vec{AP}||\vec{BP}|}$$ Now if you take negative of both vectors they cancel in the numerator and you have the same value for $\cos\theta$

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