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In all the texts about the $SL(2;\mathbb{R})$ group, people discuss the transformations of the upper half-plane.

Is it legitimate, instead, to think of $SL(2;\mathbb{R})$ as simply of $SO(1,2; \mathbb{R})$?

Then there must be a way to explicitly construct the isomorphism between them, right? If someone can provide such a construction, I would also appreciate explaining why, contrary to the $SU(2)$ vs. $SO(3;\mathbb{R})$ case, it is isomorphism but not $2\to1$ homomorphism.

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    It should be noted that the group of isometries of the half-plane is $PSL(2,\mathbb{R})=SL(2,\mathbb{R})/\{\pm I\}$, which is precisely the quotient obtained via a 2:1 map. – 2018-01-09

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It is still 2-to-1. This and several other related examples (over $\mathbb R$) are gathered at http://www.math.umn.edu/~garrett/m/v/sporadic_isogenies.pdf