It can be quite a challenge to comprehend the nuances of Bleach protagonist Ichigo Kurosaki's different forms and talents, let alone their beginnings, especially in light of how dense the series' backstory is. Ichigo's mother clearly gave him Quincy Status, but where did he receive his Quincy and Hollow skills from?
It is difficult to dissect Ichigo's family history and go further into his talents, but this explanation of his powers provides a clear, if oversimplified, account of their origins and mechanisms.
How did Quincy become Ichigo?
Ichigo's lineage allows him to draw from a power supply of Quincy, Shinigami and Hollow energy. A former Shinigami captain, Ichigo's father Isshin is the reason for his inheritance of Shinigami energy, while his mother Masaki, a pure-blood Quincy, is responsible for both his inherited Quincy and Hollow energies.
The creation and inheritance of the hybrid Shinigami-Hollow powers is explained during the Thousand Year Blood War arc, where it is revealed that Masaki was attacked and infected by a Hollow prior to Ichigo's birth. Isshin gives Masaki all of his Shinigami abilities in order to counteract the Hollow's contagious energy and save her from the creature. Masaki's soul consequently merges with Hollow and Shinigami energy, and Ichigo inherits this Shinigami-Hollow energy. Given that Quincy status is determined by blood rather than spirit, Ichigo's inheritance of Quincy power is unaffected by the influence of either Shinigami or Hollows. Ichigo is only a half-blood Quincy since after saving Masaki, Isshin was turned into a human.
Rukia's Shinigami powers were what initially awakened Ichigo's Shinigami abilities; however, because his Hollow and Shinigami abilities are combined, this also caused the awakening of his Hollow abilities. After being exposed to his Reiatsu, Ichigo's Quincy powers are reawakened during his battle with the Sternritter Quilge Opie. As a side aside, Ichigo has access to Fullbring because Fullbringers are the child of a mother who was attacked by a Hollow. The Tite Kubo-confirmed canon Can't Fear Your Own World light novels amend the requirement for Fullbringers to possession of a Soul King fragment, but as Ichigo has yet to appear in the series, it is unknown whether or not he is in possession of one, and if not, how exactly he accesses Fullbring.
How Quincy and Ichigo's Shinigami-Hollow Abilities Appear
When Ichigo enters his inner-soul to speak with his Zanpakuto spirit, it manifests itself in two forms; the man wearing glasses and a black cloak, dubbed "Old Man Zangetsu," and White Ichigo. Ichigo thought of and treated these two figures as separate beings at first, believing Old Man Zangetsu to be his true Zanpakuto spirit and White Ichigo to be a Hollow attempting to take over his soul. In actuality, Old Man Zangetsu is a manifestation of Ichigo's Quincy power, while White Ichigo is the manifestation of his hybrid Shinigami-Hollow powers.
Quincy Zangetsu hates Shinigami, and knew that the path of a Shinigami would lead to a confrontation with Yhwach and ultimately Ichigo's death. This led to Quincy Zangetsu limiting Ichigo's Shinigami-Hollow powers, working as a funnel for Ichigo's powers while at the same time masquerading as his true Zanpakuto spirit. As Ichigo's opponents grew stronger, and Quincy Zangetsu became more attached to the boy, he would eventually relinquish control over Ichigo's powers. In doing so, Quincy Zangetsu introduces Ichigo to Hollow Zangetsu, who is again, a manifestation of Ichigo's true Shinigami powers, and thus the real Zangetsu.
As Ichigo grows stronger and accesses more of the power of the real Zangetsu, he unknowingly accesses and releases more of his Hollow powers as well, evidenced by his Vizard and Vasto Lorde transformations.The effects of Ichigo using the genuine Zangetsu's strength are what Ichigo initially interprets as his inner Hollow trying to take control of his soul. Since his Hollow and Shinigami abilities are identical, Ichigo cannot use his Shinigami abilities to their fullest extent without also using his Hollow abilities. Instead of trying to possess Ichigo's soul, Hollow Zangetsu only wants to make him stronger. This is demonstrated by Hollow Zangetsu's repeated references to Ichigo as his "fellow" and his repeated entreaties to Ichigo to let him battle rather than imprison him. Quincy Zangetsu further affirms this idea, pointing out that it was always the power of a Hollow, and not his Quincy abilities, that saved Ichigo in moments of desperation.
Mastering Zangetsu: Ichigo's Zanpakuto, Explained
During his meeting with Squad Zero's Oetsu Nimaiya, the creator of all Zanpakuto, it's revealed Ichigo never truly owned a Zanpakuto. The Hollow that attacked Masaki was created by Aizen using a method similar to Zanpakuto creation, thus allowing him to act as a pseudo Asauchi for Ichigo to form his original sword. Ichigo undergoes the Asauchi trial, eventually discerning Hollow Zangetsu to be the real Zangetsu. However, following the revelation about Quincy Zangetsu, Ichigo still holds him in high esteem, and firmly asserts that both spirits were Zangetsu. As both spirits compose Ichigo's soul, and thus his Zanpakuto, Nimaiya is able to forge not one, but two Zangetsus for Ichigo: the shorter blade representing his Quincy power, the larger blade representing his Hollow and Shinigami powers.
Quincy Zangetsu and Hollow Zangetsu are two halves of the same soul, and thus two halves of the same Zanpakuto. Therefore, it is only through his achievement of balance between these powers that Ichigo maximizes his potential. This is evidenced when Ichigo allows their combined form to stab him when mastering Mugetsu, symbolizing his acceptance of both halves of Zangetsu. Against Yhwach, when he grows the Horn of Salvation, Ichigo states that he still cannot control his inner Hollow, but that his Quincy powers will help him maintain balance, finally demonstrating a true understanding of the duality of his soul and powers.