Cfrgtkky

Please be patient, this is a bit complex. I encountered some weird things when trying to answer this nice question. Start with the MWE

documentclass[tikz,border=3.14mm]{standalone}

%usetikzlibrary{intersections}

usepackage{pgfplots}

pgfplotsset{compat=1.16}

usepgfplotslibrary{fillbetween}



begin{document}

begin{tikzpicture}

 begin{axis}[axis equal]

    addplot[thick,samples=80,name path global=duck] {(-3)*x/abs(x)^(2/3)};    % From https://tex.stackexchange.com/a/144463/152550

    addplot[thick,samples=80,name path global=koala] ({sqrt(16/3)*cos((x) r)}, {sqrt(16)*sin((x) r)});

    path (0,0) coordinate (O) (1,0) coordinate(X);

    fill[red,name intersections={of=duck and koala,by={i1,i2}}]

    (i1) circle (2pt) (i2) circle (2pt);

   end{axis}

end{tikzpicture}

end{document}

So far so good. However, if one activates usetikzlibrary{intersections} and deactivates usepgfplotslibrary{fillbetween} by just moving the % down, one receives the error

 ! Package pgf Error: No shape named intersection-2 is known.

that tells us that the second intersection cannot be found. If we stick with this, but comment out [axis equal], the error disappears, i.e. the second interaction is found. What's going on here?

Even more, if one moves the determination of the intersections out of the axis environment,

documentclass[tikz,border=3.14mm]{standalone}

%usetikzlibrary{intersections}

usepackage{pgfplots}

pgfplotsset{compat=1.16}

usepgfplotslibrary{fillbetween}



begin{document}

begin{tikzpicture}

 begin{axis}[axis equal]

    addplot[thick,samples=80,name path global=duck] {(-3)*x/abs(x)^(2/3)};    % From https://tex.stackexchange.com/a/144463/152550

    addplot[thick,samples=80,name path global=koala] ({sqrt(16/3)*cos((x) r)}, {sqrt(16)*sin((x) r)});

    path (0,0) coordinate (O) (1,0) coordinate(X);

   end{axis}

    fill[red,name intersections={of=duck and koala,by={i1,i2}}]

    (i1) circle (2pt) (i2) circle (2pt);

end{tikzpicture}

end{document}

one gets

meaning that the intersections are not correct. On the other hand, it has been a common practice to compute the intersections outside the axis, see e.g. this nice answer. Notice that even if one drops axis equal in the last example the intersections are still incorrect.

The reason that coordinates for the computed coordinates do not match up is a bug in pgfplots.

The underlying reason is explained in the pgfplots manual section "4.26 TikZ Interoperability": pgfplots rescales and translates the coordinate system. Pgfplots does its best to hide this rescaling from the end user. For example, named coordinates within an axis are automatically rescaled after leaving the axis such that you can refer to them after end{axis}. This feature does not exist for the coordinates of named paths.

A workaround is to ensure that the coordinate systems match up, i.e. to disable the rescaling features of pgfplots.

This is explained in the aforementioned section of the manual. Applied to your example, this might look like

documentclass[tikz,border=3.14mm]{standalone}

%usetikzlibrary{intersections}

usepackage{pgfplots}

pgfplotsset{compat=1.16}

usepgfplotslibrary{fillbetween}



begin{document}

begin{tikzpicture}

 begin{axis}[axis equal,

    disabledatascaling,

    anchor=origin,

    x=1cm,y=1cm,

    ]

    addplot[thick,samples=80,name path=duck] {(-3)*x/abs(x)^(2/3)};

    addplot[thick,samples=80,name path=koala] ({sqrt(16/3)*cos((x) r)}, {sqrt(16)*sin((x) r)});

    path (0,0) coordinate (O) (1,0) coordinate(X);

   end{axis}

    fill[red,name intersections={of=duck and koala,by={i1,i2}}]

    (i1) circle (2pt) (i2) circle (2pt);

end{tikzpicture}

end{document}

This might affect rescaling capabilities like axis equal, though.

But again: the reason that this does not work out of the box is actually a bug: pgfplots does not translate the coordinates of named paths when it leaves end{axis}.

This is most likely because PGFPlots uses a "newer" version of the intersections library than PGF does and thus this is similar to the external library (compare https://tex.stackexchange.com/a/341662/95441). To support this you can do a diff on the both files

pgflibraryintersections.code.tex

pgfplotsoldpgfsupp_pgflibraryintersections.code.tex