Reading your question, it appears that you want to know whether a Player Character is aware of their own madness, not whether the player themself is aware of their character's madness. If that is the intent of your question, then the answer is:
Madness, as outlined in the DMG, p:258-260, encompasses a wide range of mental disorders, many of which can occur without the knowledge of the sufferer.
Disorders explicitly or implicitly listed in the madness tables include, but are not limited to:
- Obsessive compulsive disorder
- Severe anxiety
- Irrational fears/phobias
- Various states of psychosis
- Various personality disorders
- Major depression
The gold standard for classification of mental disorders is the Diagnostic and Statistical Manual of Mental Disorders (DSM-5) produced by the American Psychiatric Association. While there are numerous issues one may raise with this text, that does not preclude its usefulness.
Many mental disorders, particularly the ones that cause disturbances in personality (e.g. borderline disorder or schizophrenia) are found in the "Blind Area, Section II" of a JOHARI window - others are quite aware of the sufferer's malady but they, themselves, are not.
As an example:
One of my player's characters (lvl 13) was struck by a powerful delusion in which they truly believe they underwent a period of training in a Monk monastery and, therefore, have monk abilities. The players know of the Madness, but none of the PCs have figured it out yet. They just don't understand why the fat bard keeps trying to run across water and corner jump up walls. Hijinks!
It is very possible that a Player Character who is suffering from one or more Madness effects has no idea that they are mad.
As Acbabis notes, you should absolutely inform your players that their character is suffering from one or more Madness effects. However, for a more realistic treatment of Madness, encourage your players to RP the disorder their PC is suffering from appropriately. This may require a few minutes of research but can provide deeper and more entertaining gameplay.