How are opera singers (tenors, baritones, sopranos, etc) selected for their roles?

most singers learn about their Fach or category as they train.
( It's usually the first question a young person will ask their teacher- what voice am I?) Once they have completed their course of studies ( in university or conservatory) they can either go on to audition for agents or directly for opera company managers.
Depending on what the various theaters have programmed for their up-coming season, they will be watching out for various voice categories. Most beginning singers will take whatever is offered them for roles, which includes a whole boat-load of unimportant secondary and tertiary bit parts.
It takes an extremely exceptional beginner to jump into lead roles right away. I'd say, on the average, a singer will have 5 to 10 years of bit parts before getting a lead role in a big house. the smaller houses, who have a more throw-away approach ( like tissues, use them once and throw them away) to singers will use beginners mercilessly, often pushing them into bigger parts before their voices are quite mature enough to handle them. the poor souls are then used up and thrown away ( useless to themselves or others) to be replaced by the next batch of beginners at beginner's prices.
Yes, it does make a difference which theater the opera takes place in. voices come in different sizes as well as categories, and a voice th at has the energy and "oomph" to carry in a theater that seats 750 people may not make it at the Met or at Deutsche Oper, which seats over 1,000.
I recently read a review in the New Yorker magazine, of two people who had to jump in for ailing stars at the Met's Tristan und Isolde, the soprano in the middle of the 2nd Act.
I know this lady, and have sung with her. She is generally known throughout Europe for her sizeable instrument. the reviewer for New Yorker found her a bit undersized for the role...
the theater I work in seats 850 people. It's still considered a small theater. Some of our soloists, who came as beginners, moved on to make careers in more sizeable houses, and even to the Met. Some were used up.

opera singers audition for managers, who find them auditions in all sorts of opera companies across the land. every opera singer is a "fach" (a source of great controversy, change, and drama for some reason) or a "box" of their voice type, and even more specifically, what repitoire they specialize in. for example, a mezzo (her fach- the middle female voice) may specialize in mozart pants roles (where a girl plays a young boy) and other light roles. or a mezzo may specialize in heavy dramatic repitoire. maria callas was well known for singing all sorts of things outside of her fach.

no I thought they are selected by themselves