Sukhoi and Tupolev were design bureaus (OKBs, from the Russian acronym), two among many others. As was common Soviet practice, they were set up with numbers and later acquired semi-official names. Since the OKBs were set up and centered around a chief designer, either a prominent or promising man, they often acquired his name, and if that name were famous enough they would keep it after he was gone. Sukhoi, OKB-51, was led by Pavel Sukhoi essentially until his death in 1974... except for a period in which Stalin forced him to work under Tupolev. That arrangement ended with Stalin's death. It kept his name even through various successor corporations and divisions. Andre Tupolev was in charge of OKB-156, except for some years where he was in prison (designing aircraft) after a Stalinist purge. OKB Tupolev not only kept the name, but was run by his son until he died in 2001.

The design bureaus were small organizations dedicated to design of technological (usually military) products. Those specializing in aircraft would design, prototype, compete, and help set up production of their aircraft. They were not involved in serial production of those aircraft; that task was assigned to one or more plants that (usually) had experience in aircraft production. OKBs did tend to have close relationships with certain factories; OKB Ilyushin was located at Zavod (literally: Factory) 30 and used its facilities to build prototypes. The Moscow Air Production Organization (MAPO), largely at Zavod 1, was closely associated with OKB Mikoyan and Gurevich (MiG). Most plants would build a variety of designs at the same time. The production history is complicated, especially with some factories moved during WWII.

Several OKBs would design aircraft for a needed role, not unlike the usual Western practice, both for competition and in case a design failed. Under Stalin, you did not want your design to fail; in later times designs and designers became much more bureaucratically secure. The lack of intellectual property protections within the Soviet economy meant that design features would freely migrate (or be stolen, depending on your perspective) between design bureaus, and some aircraft competitions would involve mergers of the competing designs. Since designs would be expected to be manufactured in various locations with various levels of experience, designs tended to emphasize ease of manufacture, and make extensive use of existing or common components.

I don't have a source on the internal operations of an OKB, but one would expect personnel decisions to work like other Soviet organizations. The answers by /u/Minardi-Man in Getting a job in the USSR may be instructive. Several OKBs did manage to control succession of the head designer.

In the post-Soviet era, many successful OKBs became manufacturing corporations in the Western style by merger with one of the factories. MAPO and MiG, for example, became the modern MiG. However, they were still subject to various government reorganizations, some of which are too recent to be discussed here.

If interested in more details, you might look at "Soviet Aircraft Industry" by Peter G Dancey, much of which can be read online.