Fuse rating and safety
The "speed", rupture capability, and voltage spec of a fuse are each separate traits that aren't necessarily related.  They will be specified according to the nature of the application.

Many cheap multimeters (i.e. &lt;50$us) will only have glass fuses, because they're cheaper.  But 'high rupture' capability (the ability to break a circuit up to a rated voltage, when a high fault current is flowing) is important in a multimeter (and in a circuit breaker), especially in a DC scenario, because DC >40V can easily form an arc and continue the current flow, even though the fuse has blown.

For the "20A" fuse in your meter, a HRC type would be wise, and rated for the max V rating of your multimeter, and same speed as the original.