The Cauchy product theorem for infinite series of complex numbers states if $\sum a_n$ and $\sum b_n$ are two absolutely convergent series then the Cauchy product $\sum c_n$, where $c_n=\sum_{p+q=n} a_pb_q$ is also absolutely convergent and $\sum c_n=(\sum a_n)(\sum b_n)$.
I was wondering if there is a counter example if we weaken the conditions, specifically:
Do there exist series $\sum a_n$ and $\sum b_n$ which are both convergent but not both absolutely convergent, such that the Cauchy product $\sum c_n$ converges, but $\sum c_n \neq(\sum a_n)(\sum b_n)$?
No, a counterexample satisfying the property you seek cannot exist, which we can see by Cesàro's theorem.
For a series $\sum d_n$, the Cesàro sum is the limit of the average of the partial sums:
$$\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N\sum_{i=1}^nd_i$$
If a sum is classically summable, then it is also Cesàro summable, and its Cesàro sum matches its classical sum.
Now Cesàro's theorem says that in general if $\sum a_n$ and $\sum b_n$ are conditionally convergent, then the Cauchy product will be Cesàro summable, and its Cesàro sum will be $(\sum a_n)(\sum b_n),$ even though it need not be classically convergent to $(\sum a_n)(\sum b_n).$
Hence what you are asking for cannot happen. For $c_n=\sum_{p+q=n} a_pb_q$, either $\sum c_n$ is not classically convergent, but it is Cesàro summable and its Cesàro sum is $(\sum a_n)(\sum b_n).$
Or else it is classically convergent, in which case it must still converge to its Cesàro sum $(\sum a_n)(\sum b_n).$