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Partial (non-)answer, to be completed if divine inspiration strikes ;-)...

Solving this system in SR doesn't seem to be possible.

sage: var("a, b, c, d")
(a, b, c, d)
sage: eq1 = c==a*(a/(a+2*b))+2*b*(a/(a+b)) ; eq2 = d==a*(b/(a+2*b))+b*(b/(a+b))

Solving the first equation for b is trivial, leading to two solutions

sage: Solb1 = solve(eq1, b, solution_dict=True)
sage: Solb1
[{b: -1/4*(3*a^2 - 3*a*c + sqrt(-7*a^2 + 6*a*c + c^2)*a)/(2*a - c)},
 {b: -1/4*(3*a^2 - 3*a*c - sqrt(-7*a^2 + 6*a*c + c^2)*a)/(2*a - c)}]

Substituting this solution in the second equation leads to non-linear equations, e. g. :

sage: eq3 = eq2.subs(Solb1[0])
sage: eq3
d == -1/2*(3*a^2 - 3*a*c + sqrt(-7*a^2 + 6*a*c + c^2)*a)*a/((2*a - c)*(2*a - (3*a^2 - 3*a*c + sqrt(-7*a^2 + 6*a*c + c^2)*a)/(2*a - c))) + 1/4*(3*a^2 - 3*a*c + sqrt(-7*a^2 + 6*a*c + c^2)*a)^2/((4*a - (3*a^2 - 3*a*c + sqrt(-7*a^2 + 6*a*c + c^2)*a)/(2*a - c))*(2*a - c)^2)

This equation cannot be solved by Maxima's solver : calling it without extra arguments fails :

sage: Solam = solve(eq3, a, algorithm="maxima", solution_dict=True) # Apparent success...
sage: len(Solam)
4

This solution isn't a solution :

sage: [u[a].variables() for u in Solam]
[(), (a, c, d), (a, c, d), (a, c, d)]

The three last "solutions" are just rewritings of eq3 ; the last is a triviality :

sage: Solam[0]
{a: 0}

leading to `{b:0} when substituting in the first equation, unacceptable as noted before...

Calling

sage: Solamp = solve(eq3, a, algorithm="maxima", solution_dict=True, to_poly_solve=True)

didnt't return after > 30 minutes

Similarly, algorithm="sympy" didn't return after > 30 minutes. Trying giac fails (Giac's solver seems limited to polynomials...). Frica's solver gives an answer seeminlgly using intermediary computations, which do not seem to be capturable "simply" by Sage (to be explored further...).

Another possibility is to use Sage's apparatus for polynomials :

sage: R1.<c0, c1>=QQbar[]
sage: R2.<c2, c3>=FractionField(R1)[]
sage: J=R2.ideal([R2(u.subs(dict(zip((a, b, c, d), (c0,c1, c2, c3))))) for u in (eq1, eq2)])

This ideal, which is, in some sense, the "best" solution to your problem, can be defined in Sage

sage: J
Ideal (c2 + (-c0^3 + (-3)*c0^2*c1 + (-4)*c0*c1^2)/(c0^2 + 3*c0*c1 + 2*c1^2), c3 + (-c0^2*c1 + (-2)*c0*c1^2 + (-2)*c1^3)/(c0^2 + 3*c0*c1 + 2*c1^2)) of Multivariate Polynomial Ring in c2, c3 over Fraction Field of Multivariate Polynomial Ring in c0, c1 over Algebraic Field

One can even be sure that this solution is a finite set :

sage: J.dimension()
verbose 0 (3998: multi_polynomial_ideal.py, groebner_basis) Warning: falling back to very slow toy implementation.
verbose 0 (1082: multi_polynomial_ideal.py, dimension) Warning: falling back to very slow toy implementation.
0

Alas :

sage: J.variety()

[ Snip... ]

NotImplementedError: root finding for this polynomial not implemented

OTOH, I'm far from an expert in this domain, and might have gone in completely wrong direction...

My tentative concusion isn't that Sage cannot solve this system, but rather that I can't solve this system with Sage but by using its mathematica interface to the Wolfram engine...