Does this paper imply graph isomorphism is polynomial for cubic and $4$-regular graphs?
Their paper doesn't say anything about general regular graphs. Note that the $b$ is constant. Actually all of their excluded subgraphs are constant. Their polynomial time algorithm is consist of considering all possible $K_{2b+1}$'s in that case. This causes to running time $\Omega(n^{2b+1})$. Means if we want to use their algorithm for $r$-regular graph then we should use running time $\Omega(n^{2r+1})$, means when $r$ is not fix their algorithm is not polynomial time in the size of input. Actually we can say that they proved the problem in that case belongs to XP when the problem parametrized by $b$. Don't mix parametrized complexity with usual polynomial time algorithms.