Why are there 3.15A fuses?
Each fuse rating is about 1.26 x higher than the previous value. Having said that preferred values do tend to be located at slightly easier to remember numbers: -


100 mA to 125 mA has a ratio of 1.25
125 mA to 160 mA has a ratio of 1.28
160 mA to 200 mA has a ratio of 1.25
200 mA to 250 mA has a ratio of 1.25
250 mA to 315 mA has a ratio of 1.26
315 mA to 400 mA has a ratio of 1.27
400 mA to 500 mA has a ratio of 1.25
500 mA to 630 mA has a ratio of 1.26
630 mA to 800 mA has a ratio of 1.27
800 mA to 1000 mA has a ratio of 1.25


315 mA just happens to span quite a large gap between 250 mA and 400 mA so I suppose the ratio-halfway point should really be \$\sqrt{250\times 400}\$ = 316.2 mA. Near enough!

But, the bottom line is that consecutive fuses (in the standard range shown above) are "spaced" \$10^{1/10}\$ in ratio or 1.2589:1. See this picture below taken from this wiki page on preferred numbers: -



These numbers are not-unheard of in audio circles either. The 3rd octave graphic equalizer: -



See also this question about why the number "47" is popular for resistors and capacitors.


  Is it even possible to make fuses with better than +/-5% tolerance?


I expect it is but fuses don't dictate performance only functionality so, tight tolerances are not really needed. Resistors on the other hand totally dictate performance on some analogue circuits so tight tolerances (down to 0.01%) are definitely needed.