I know there were still other competitors like Amiga, but it seems that by 1993 the market had generally settled on those two platforms. Why did the other companies like Atari, Commodore, etc fail to unseat Microsoft and Apple?
Apple were far from dominant in 1993 and were in a stagnation or decline until the return of Jobs in the late '90s. They had a niche targeting publishing and other creative industries.
Microsoft benefitted from the almost accidental decision of IBM to use an operating system provided by MS. Prior to that, Microsoft sold development tools. When the planned vendor for the operating system failed to grasp the opportunity of a deal with IBM, Microsoft bought an OS to fill the gap. There is a quite a lot written about this deal e.g. Bill Gates had a family connection to the head of IBM.
The giant IBM's entry into the microcomputer business helped to lend credibility to the market, which was seen as immature and hobbyist focused. The terms they agreed with Microsoft allowed Microsoft to sell MS-DOS to others (a mistake). Compaq and other PC manufacturers started making IBM compatible 'clones' and Microsoft provided the software. This combination became a de facto standard for business desktop machines.
Microsoft built on their good fortune and competed aggressively to grow their market share, by acquiring and developing products in different categories. When IBM planned to replace DOS with a newer technology, and have Microsoft write it with them, Microsoft was also working on behind the scenes the first version of Windows that would be highly successful (Windows 3.0). This stymied IBM's plan to regain leadership of the PC platform.
Microsoft under Gates was very aggressive and seen by many in the industry as ruthless or unethical. Either way, they achieved a 90% marketshare, which meant that most software was developed for their OS and created a virtuous circle from a Microsoft perspective.