The name "x86" was never 'given' or 'designed' this way. If I remember correctly, it more or less evolved as a convenient abbreviation for a whole range of compatible processors.
Back in the day when PC's became popular, it was important that your PC was "IBM Compatible". This meant, among other things, your PC must have an Intel 8086 or an 8088. Later, when Intel released more powerful processors such as the (rare) 80186 or (popular) 80286, it was still important that your PC was just "MS-DOS" or "IBM Compatible". The 80286 was just a faster processor. It had a protected mode feature, but little software actually used or even required that.
The next step was the 80386. This was an improvement over the 80286 because it had a mode that provided full backward compatibility with 8086 programs. Operating systems such as OS/2, DesqView and MS-Windows used this mode to provide backward compatibility with existing software. Other operating systems such as Linux and *BSD's designed for PC hardware also depended on some new features of the 80386 without actually providing direct compatibility with existing MS-DOS software. All these systems required a 80386 processor.
Then came the 80486. An even faster and more powerful processor but mainly backward compatible with the '386. So if you bought a '486 you could still run software designed for the '386. The package would say 'needs a 386 or better' or 'needs 386 or 486'
Along came the 80586 or Pentium. And then the Pentium Pro, also known as 80686...
By this time software developers got tired of listing all possible numbers and since most software was still written to be able to run on a '386, the whole list of numbers was abbreviated to just "x86". This later became synonymous with "32 bit", because the 80386 was a 32 bit processor and hence software that's written for 'x86' is 32-bit software.