Same style with gnuplot than with normal graph with TikZ

I propose the solution below which does not use gnuplot. I hope you are not unconditionally in love with it.

It uses TikZ only and a parametrization of the singular cubic.

The parametrization is obtained by projecting the curve from the origin onto the line x+y=1. We get (x, y) = 6t/(1+t^3)(1, t).

We have to make some choices during the drawing process since t is different from -1. This is the reason for the four draw commands. They might be transformed into two though.

Your axes are too small for the coefficient 6 in the cubic's equation. So, I scaled down the curve to fit the interesting part in the desired rectangle.

documentclass[11pt, border=.5cm]{standalone}

usepackage{tikz}
usetikzlibrary{calc, math}

begin{document}

tikzmath{%
  real sizeGraph;
  sizeGraph = 1.4;
}
begin{tikzpicture}  
  draw[domain=-0.91:0.91, smooth, variable=x, red, ultra thick]
  plot ({x}, {sqrt(1-x*x)});
  draw[domain=-1:-0.9, smooth, variable=x, red, ultra thick]
  plot ({x}, {sqrt(1-x*x)});
  draw[domain=0.9:1, smooth, variable=x, red, ultra thick]
  plot ({x}, {sqrt(1-x*x)});

  draw[domain=-0.91:0.91, smooth, variable=x, red, ultra thick]
  plot ({x}, {-sqrt(1-x*x)});
  draw[domain=-1:-0.9, smooth, variable=x, red, ultra thick]
  plot ({x}, {-sqrt(1-x*x)});
  draw[domain=0.9:1, smooth, variable=x, red, ultra thick]
  plot ({x}, {-sqrt(1-x*x)});
  draw[->] (-sizeGraph,0) -- (sizeGraph,0) node[right] {$x$};
  draw[->] (0,-sizeGraph) -- (0,sizeGraph) node[above] {$y$};
  
  node[below=1.5cm, align=flush center] {$V(X^2+Y^2-1)$};
end{tikzpicture}
qquad
begin{tikzpicture}
  draw[samples=1000, domain=-sizeGraph:sizeGraph, smooth,
  variable=x, blue, ultra thick] plot ({x}, {x*x});
  draw[->] (-sizeGraph,0) -- (sizeGraph,0) node[right] {$x$};
  draw[->] (0,-sizeGraph) -- (0,sizeGraph) node[above] {$y$};
    
  node [below=1.5cm, align=flush center]{$V(Y-X^2)$};
end{tikzpicture}
qquad
begin{tikzpicture}
  draw[samples=1000, domain=-sizeGraph:sizeGraph, smooth,
  variable=x, orange!60!black, ultra thick] plot ({x}, {x});
  draw[samples=1000, domain=-sizeGraph:sizeGraph, smooth,
  variable=x, orange!60!black, ultra thick] plot ({x}, {-x});
  draw[->] (-sizeGraph,0) -- (sizeGraph,0) node[right] {$x$};
  draw[->] (0,-sizeGraph) -- (0,sizeGraph) node[above] {$y$};
  
  node [below=1.5cm, align=flush center] {$V(Y^2-X^2)$};
end{tikzpicture}
quad
tikzmath{%
  integer N{-}, N{+}, j;
  N{-} = 21;
  N{+} = 22;
}
begin{tikzpicture}
  begin{scope}[red, ultra thick, scale=.4]
    draw (0, 0)
    foreach i [evaluate=i as j using i/20] in {1, ..., N{+}}{%
      -- (${1/(1+j^3)*(6*j)}*(1, j)$)
    };
    draw (0, 0)
    foreach i [evaluate=i as j using -i/40] in {1, ..., N{-}}{%
      -- (${6*j/(1+j^3)}*(1, j)$)
    };
    
    draw (0, 0)
    foreach i [evaluate=i as j using i/20] in {1, ..., N{+}}{%
      -- (${1/(1+j^3)*(6*j)}*(j, 1)$)
    };
    draw (0, 0)
    foreach i [evaluate=i as j using -i/40] in {1, ..., N{-}}{%
      -- (${6*j/(1+j^3)}*(j, 1)$)
    };  
  end{scope}
    draw[->] (-sizeGraph,0) -- (sizeGraph,0) node[right] {$x$};
  draw[->] (0,-sizeGraph) -- (0,sizeGraph) node[above] {$y$};

  node [below=1.5cm, align=flush center] {$V(X^3+Y^3-6XY)$};
end{tikzpicture}

end{document}