A person is considered an adult and held responsible for religious and legal obligations on attaining puberty, as evidenced by the Quranic verses and ahadith that condition such obligations on having reached puberty, and the hadith about the three cases from whom the pen is lifted and their actions are not recorded:
وإذا بلغ الأطفال منكم الحلم فليستأذنوا
And when the children among you reach puberty, let them ask permission
— Quran 24:59
وابتلوا اليتامى حتى إذا بلغوا النكاح فإن آنستم منهم رشدا فادفعوا إليهم أموالهم
And test the orphans [in their abilities] until they reach marriageable age. Then if you perceive in them sound judgement, release their property to them
— Quran 4:6
رفع القلم عن ثلاثة عن النائم حتى يستيقظ وعن الصبي حتى يحتلم وعن المجنون حتى يعقل
The pen has been lifted from three: a sleeper till he awakes, a boy till he reaches puberty, and a lunatic till he comes to reason.
— Abu Dawud
غسل يوم الجمعة واجب على كل محتلم
The taking of a bath on Friday is compulsory for every male (Muslim) who has attained the age of puberty.
لا يقبل الله صلاة حائض إلا بخمار
The Salat of a women who has reached the age of menstruation is not accepted without a Khimar
— Jami at-Tirmidhi
Reaching a numerical age does not make one an adult, except where puberty is early or delayed. Hence a maximum age is defined after which the person is considered an adult even if the signs of puberty are not evident by then, similarly a minimum age is defined so a person is not considered an adult even if the signs are evident before that.
There are different schools of thought on the maximum age such as 15 years according to the Shafis and Hanbalis or 18 years according to the Hanafis and Malikis. Possibly the stronger of these is 15 years based on the hadith:
قال حدثني ابن عمر ـ رضى الله عنهما ـ أن رسول الله صلى الله عليه وسلم عرضه يوم أحد وهو ابن أربع عشرة سنة، فلم يجزني، ثم عرضني يوم الخندق وأنا ابن خمس عشرة فأجازني
Narrated Ibn Umar: Allah's Messenger (ﷺ) called me to present myself in front of him or the eve of the battle of Uhud, while I was fourteen years of age at that time, and he did not allow me to take part in that battle, but he called me in front of him on the eve of the battle of the Trench when I was fifteen years old, and he allowed me (to join the battle)."
— Sahih Bukhari
Regarding Zakat there is a difference of opinion among the scholars. The Shafis, Malikis and Hanbalis hold the view that Zakat should be given from a minor's property by their guardian. The Hanafis hold the view that Zakat is not payable from a minor's wealth. (source Islamic Jurisprudence According To The Four Sunni Schools , Hanafi view in Hidayah).