Torsion point

The tangent line through $P$ has the equation $y=2x-1,$ so the other point of intersection is $(0,-1).$ The additive inverse of $(0,-1)$ is the point of intersection of the vertical line through $(0,-1)$ with the curve, which is $Q = (0,1).$ So $2P = (0,1).$

Now $3P = 2P+P = Q+P$ is computed by looking at the line $y=x+1$ through them both, and finding the third point of intersection. This is clearly $R = (-1,0).$ Now the additive inverse of $R$ is $R,$ since the vertical line through $R$ is tangent to the curve there. So $3P = -R = R.$ Notice that this means that $2R$ is the identity, so $6P = 2R = O$ (here $O$ is not the origin, but the point at infinity, which is the identity element of the group of torsion points.) This shows that $P$ is a $6$ -torsion point.

Now Lagrange's theorem implies that the smallest positive integer $n$ for which $P$ is an $n$ -torsion point is $1,2,3,$ or $6,$ and we've seen that it's not $1,2,$ or $3,$ so it must be $6.$

Alternatively, if you don't want to use group theory, for completeness' sake, we can write down all the multiples of $P.$ $\begin{aligned} P &= (2,3) \\ 2P &= (0,1) \\ 3P &= (-1,0) \\ 4P &= (0,-1) \\ 5P &= (2,-3) \\ 6P &= O \end{aligned}$