Your meter does drop some voltage when its measuring current, it needs too so it can actually measure current, however the fuse itself should only drop a tiny voltage.
The voltage rating as mentioned means if the fuse 'blows' the circuit will be reliably disconnected up to the voltage rating of the fuse, if you over voltage the fuse you might not notice anything until the fuse fails, then the would be a risk that the fuse would either not fail at all or arc over internally, neither of which would protect you or the circuit.
For a meter its critically important for your own safety to use the correct fuse, you hold the probes & leads and maybe the meter itself while your connected to power, it wouldnt be nice for the leads & meter to vaporize while your holding them, meter fuses are expensive but necessary.