Ministry is used in at least four incompatible senses and most confusion about it comes from sliding between them.
An office. A minister is the ordained pastor of a congregation, appointed by a defined process, usually paid, with authority to preside at worship and the sacraments that church recognises.
A class. The ministry means the whole ordained body of a church, as in entering the ministry.
An activity. A prison ministry, a music ministry, a youth ministry. This sense is open to anyone, requires no ordination, and is how most people in most congregations use the word about themselves.
A gift. In Quaker usage, and in a number of charismatic and Pentecostal churches, ministry is something the Spirit gives to a particular person for a particular occasion. It is not conferred by an institution, cannot be withheld by one, and may be exercised without permission. A Quaker meeting recognising a gift in the ministry is acknowledging a fact, not creating a status.
The first and last senses describe arrangements that cannot both be right about where authority comes from, and a great deal of ecumenical discussion has foundered on participants using the same word for both. Agreements on the mutual recognition of ministries have to specify which sense is meant before they can say anything at all.
The New Testament vocabulary does not settle it. The earliest Christian communities show apostles, prophets, teachers, overseers, elders and servants without a stable structure, and every subsequent reform movement has appealed to that period against whatever arrangement it wished to overturn. What the last two centuries added, and what none of the ancient vocabulary carries, is the expectation of a salaried profession with training, credentials, a career and a pension.