Let $f,g: \mathbf{R} \to \mathbf{R}$ be two strictly increasing continuous functions such that, for all $a,b,c,d\in \mathbf{R}$, it holds $$ f(a)+f(b)=f(c)+f(d) $$ if and only if $$ \forall h \in \mathbf{R},\,\, g(a+h)+g(b+h)=g(c+h)+g(d+h). $$
Is it true that $f$ and $g$ have to be necessarily (both) linear?
(Clearly, it can be assumed without loss of generality that $f(0)=g(0)=0$ and $f(1)=g(1)=1$. Hence the claim can be rewritten as $f(t)=g(t)=t$ for all $t$.)
No, it is not true.
Let $f(x) \equiv \alpha e^x$, $g(x) \equiv \beta e^x$, where $\alpha$ and $\beta$ are some positive constants. Then $$f(a) + f(b) = f(c) + f(d) \iff e^a + e^b = e^c + e^d \\ \iff \forall h \in \mathbb R \, \left(e^{a+h} + e^{b+h} = e^{c+h} + e^{d+h}\right) \\\iff \forall h \in \mathbb R \, \left(g(a+h) + g(b+h) = g(c+h) + g(d+h)\right).$$