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Here is a partial answer (I hope somebody can come with a better one!):

Plotting the solutions of an equation in $\mathbb R^3$ can be done using the method implicit_plot3d. So you can visualize the solutions of both equations as follows:

sage: var('x,y,z')
sage: s1 = implicit_plot3d((x+y)*(x-z^3), (x,-2,2),(y,-2,2), (z,-2,2))
sage: s2 = implicit_plot3d(x*y+y^2, (x,-2,2),(y,-2,2), (z,-2,2), color="red")
sage: show(s1)
[solutions of the first equation]
sage: show(s2)
[solutions of the second equation]

You can also visualize both solution sets together:

sage: show(s1+s2)

Since you are looking for solutions in $\mathbb R^3$, having both equations equal zero is the same as the sum of their squares equal zero. So in principle you could do

sage: implicit_plot3d(((x+y)*(x-z^3))^2+(x*y+y^2)^2, (x,-2,2),(y,-2,2), (z,-2,2))

But the problem is that if you try this you will see an empty set of solutions. I am not sure about the reason.

Finally, even though it is not visualization, note that you can have also the set of solutions using solve:

sage: sol = solve([(x+y)*(x-z^3),x*y+y^2], [x,y,z])
sage: sol
[[x == r1, y == -r1, z == r2], [x == 0, y == 0, z == r3], [x == r4^3, y == 0, z == r4]]