Polynomials generating the same $p$-adic fields

This is false. For example, $X^2+2$ and $X^2+6$ are both equal and irreducible mod $4$.

However, since $\mathbb Q_2$ does not contain a square root of $3$, their roots give different extensions of $\mathbb Q_2$.

Check out Lang's Algebraic Number Theory section II.2. In a nutshell, if two polynomials are $p$-adically close then their roots are close as well, and by Krasner's lemma the fields they generate over $\mathbb Q_p$ will be the same.