The proper collective term is a “religious institute”. Not all institutes are orders. The last order to be founded was the Jesuits by special permission; no new religious orders can be founded due to a ban by ecumenical council. However, lots of religious institutes have been founded since then. The Code of Canon Law defines different types and how they are organized. The SOLT is a Society of Apostolic Life. This resembles a religious institute, but differs in minor ways.
The defining characteristic of a religious institute is not especially which saint(s) they admire, but rather their charism. This is a specific type of spiritual gift which is unique to the institute and is usually well-defined in their rule or constitution.