Some history on this feature: it started as a userscript written by Scott Morrison with contributions from Physics moderator Manishearth, proposed and developed on this meta.MO thread, which was then officially integrated by the SE devs into MathOverflow, and subsequently also enabled on math.se.
In principle, if we want this to be implemented on this site, it should just mean flipping a few switches dev-side. However, it's not clear to me what back-end the script is using for the search function, but if it's using things like math ZentralBlatt or MathSciNet then it is entirely plausible that the search will not have fantastic coverage over physics - and indeed it seems that it doesn't; a quick search for this paper on the math.se tool didn't find it by title, which is worrisome.
Luckily, though, if individual users want to have this, it should be plenty easy to just install the userscript (i.e. just copy this javascript file into Tampermonkey/Greasemonkey/your userscript extension of choice). As such, I propose that we gather some reasonably-sized population of users on this site, spanning as wide a range of physics subfields as possible, to take the search back-end for a test drive and see whether it actually works for physics papers in real-world usage.
Edit: as pointed out in the comments, and documented in this meta.MO thread, it appears that the math.se+MO citation search button does use a ZentralBlatt search back end, which will severely hinder its effectiveness for physics papers. The back-end should in principle be relatively independent of the front-end, so in principle it should be possible to rewrite it to use e.g. the Bielefeld Academic Search Engine or some other suitable API (which should be possible since oaDOI use BASE as their back end, and it's open sourced so we can probably learn from their calls), but it will require nontrivial adaptations.
So, if people really want this, the first thing to do is to look for a suitable back end, as proposed in Looking for a back end for a Physics SE literature citation button.
And, while we're here, I would like to propose a change to the citation formatting to make it mesh better with the standards in physics, from
Carb\'o-Dorca, Ramon, On Einstein-Podolsky-Rosen paradox, J. Math. Chem. 41, No. 3, 209-215 (2007). ZBL1119.81009.
to something like
R. Carbó-Dorca. On Einstein-Podolsky-Rosen paradox, J. Math. Chem. 41 no. 3, 209-215 (2007).
where in particular we should ditch the ZentralBlatt references, which are only useful to mathematicians, and add arXiv references whenever possible (which e.g. the second result in your search does not include in the citation output, even though it exists). And, if we can get the accented characters like \'o and \v{c} and the like working, it'd be nice.