While a DID system can include a lot of inside people with different ages, genders, personalities and ailments, the mental health disorder describes the whole system. But, as tale said, different alters can manifest different mental health symptoms, such as depression, anxiety, paranoia, hallucinations, psychosis and so on. These are some of the same symptoms that other mental health disorders manifest, and in general DID is comorbid with other mental illnesses, such as BPD, CPTSD, ADHD, other personality and mood disorders and so on. This is because growing up in extreme emotional/physical/sexual abuse, with no ability to escape, can result on one's mind being fractured to provide a certain level of escape through extreme dissociation (hence DID). You can imagine that someone who survived that in early childhood and on will have a host of mental health issues, some of which are sometimes possible to compensate for, or to medicate for, but most are impossible to cure, since brain plasticity dramatically declines after childhood.
TL;DR: specific symptoms, shared disorders.