Flaw in linear programming solution for multi-commodity flow problem?
The linear programming formulation from CLRS is fine.  There is no problem with it.

Your purported counterexample isn't a valid counterexample -- it  doesn't satisfy all of the inequalities.  Take a closer look at the specification of the second inequality.  For instance, when we are dealing with commodity $a$ ($i=a$) it says that we should have

$$\sum_{v \in V} f_{auv} - \sum_{v \in V} f_{avu} = 0$$

for all $u \in V \setminus \{s_a,t_a\}$.  In particular, consider the case $u = t_b$.  We have

$$\sum_{v \in V} f_{at_bv} - \sum_{v \in V} f_{avt_b} = 0 - 2 \ne 0,$$

so the inequality is violated.

To put it in English: the second inequality requires conservation of flow for each commodity, except at that commodity's source and sink.  For example, for commodity $a$, we must have conservation of flow at every vertex except $s_a$ and $t_a$.  In your proposed counterexample, conservation of flow for commodity $a$ is violated at vertices $s_b$ and $t_b$; that's not allowed.