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The documentation of the solve function states:

Algebraically solve an equation or system of equations (over the complex numbers) for given variables. Inequalities and systems of inequalities are also supported.

So it should not be expected to work in other cases.

The function desolve_system can be used for first-order systems of ordinary differential equations:

sage: desolve_system([eq1,eq2],[f,g])
[f(r) == -r^2 + f(0), g(r) == g(0)]

The documentation of the solve function states:

Algebraically solve an equation or system of equations (over the complex numbers) for given variables. Inequalities and systems of inequalities are also supported.

So it should not be expected to work in other cases.

The function desolve_system can be used for first-order systems of first-order ordinary differential equations:

sage: desolve_system([eq1,eq2],[f,g])
[f(r) == -r^2 + f(0), g(r) == g(0)]

The documentation of the solve function states:

Algebraically solve an equation or system of equations (over the complex numbers) for given variables. Inequalities and systems of inequalities are also supported.

So it should not be expected to work in other cases.

The function desolve_system can be used for systems of first-order ordinary differential equations:

sage: desolve_system([eq1,eq2],[f,g])
[f(r) == -r^2 5*r + f(0), g(r) == r + g(0)]