Subjective: based on or influenced by personal feelings, tastes, or opinions.
Objective: (of a person or their judgment) not influenced by personal feelings or opinions in considering and representing facts.
What is the difference between being schizophrenic and having the "opinion" that Satan speaks to you through your neighbor's cat?
Mental illness is defined by what a person feels and believes.
What you feel and believe is subjective but you can objectively establish that a particular cluster of subjective feelings and beliefs is abnormal vis-a-vis a populate.
Then when you ask people the same questions and their cluster of responses fits the abnormal definition you can objectively say that they have a mental illness.
The objectivity does not come from the doctor observations or patient self-survey, it comes from the study that created the questions and proved that a certain score or cluster of responses is statistically abnormal.
Honestly, a lot of the time it comes down to observable impact. If a patient comes to a doctor and says that they feel depressed, that's one thing, if they have lost their third job this year because they can't do anything but lie in bed and seem to not care about their own lives (especially if they used to not be like this in the past) that's something else.
How do you objectively verify "manic"? You can track hours slept. No one fakes sleeping two hours a day for a month while having, say, good motor skills in the daytime.
This matters because recurring brief hospitalization is necessary for some people to get back on their axis. This is NOT "sanity days" or "mental healing", it's a clear cut medical issue.