Before engine-compatible assets were good enough to render in-engine, cinematics that involved character animation were authored with an entirely parallel path, using the same reference art, in a character animation program. Maya and 3d Studio Max were popular at the time and most reference art and lookdev was done with them anyway, so that was a popular choice, and Softimage (RIP) was an artist favorite in offline animation too.

Sometimes making the cinematics wasn't a core competence of the studio working on the game, so VFX or animation studios would be contracted to do this. Often this meant the studio was already set up to work for TV or Film, so it was staffed such that a 3D render would reach a compositor relatively quickly, where many manual corrections would happen in 2D space. Compositing software has largely consolidated nowadays to the few survivors (Nuke, Flame mostly) but at the time there were many, like Combustion and Henry (Quantel), and even After Effects was used a bunch in places I worked.

To this day, not all engines love creating cinematics, because even if an external/offline renderer can do the full render (like in Unreal Engine), some engines don't support the animation systems required to do cinematics, or don't support them at a velocity artists like. In other words, the same software workflows are used today.