Is there a higher risk of having a mental illness?

Yes, but...: If you have a family history of mental illness, there is a greater likelihood than at random in the population, but even for identical twins, the risk of schizophrenia if one has the illness is only 50% for the other twin, for bipolar disorder it's 40-70% risk, and for major depression it's 50%. So even with identical dna there is still only about 50% contribution of inheritance by family biology.