So my question really is: does the fuse protect the meter from damage, or is it more to prevent fires and thus protect the user?
Without making any assumption about how the fuse is blown, both scenarios are theoretically possible. It would require a case-by-case analysis to determine the consequence of a particular misuse. For serious lab work, it may be necessary to send the multimeter back to the calibration department for a recheck, since it's the policy of many labs to require re-calibration after an instrument is repaired - replacing a fuse may count as a "repair" as well.
That being said, I believe it's unlikely to damage a multimeter by a mild overcurrent, in particular, the chance of damaging expensive active components (such as its analog-to-digital converter) is extremely low.
To see why, let's take a look at the schematic of a representative multimeter, the Fluke 87. Other multimeters have similar protections in place.
When the multimeter probe is plugged into the "milliamp" and "common" sockets and set to the "milliamp" range, the measured electrical current flows across a 1A fast-blow fuse, to the mA range switch, to a 0.99 Ω and a 0.01 Ω resistor, and finally leaves the meter via the "common" terminal. At the top of the 0.99 Ω resistor, an "amps sense" line connects it to the pin "AP3" of the multimeter ASIC (which has an integrated analog-to-digital converter).
How does the input circuit work? A multimeter measures current by sensing the voltage across a known low-value (e.g. 1 Ω in this case) shunt resistor, and uses Ohm's law V = IR to determine the current. What are the consequences if there's an overcurrent?
Overvoltage. Due to Ohm's law V = IR, a slightly higher voltage drop appears across the shunt resistor. With an ideal low-voltage source, or a current-limited voltage source, this is unlikely to cause damage to the active components.
If it's connected to a "high" voltage source but it's current-limited (by series resistance, by active regulation, by capacitance of a charged capacitor). it's unlike to cause damage, due to the low shunt resistance value used, the voltage drop across the shunt would be small. It may also force the voltage source to discharge or drop quickly.
If it's connected to a low voltage source around a few volts without current limit, it's unlike to cause overvoltage damage to the active electronics in the meter. Suppose the multimeter is connected directly across a 5 V voltage source, the short-circuit current is 5 A, and the voltage across the two resistors is 5 V.
If it's connected across a "high" voltage, high current source, there can be a serious overvoltage.
Overheat. Due to Joule's law P = I²R, increased power is dissipated into the shunt resistors. In case the short-circuit current is 5 A, the instantaneous power dissipation into both resistors would be 25 W. If the overcurrent is persistent, heat can cause permanent damage to this (inexpensive) shunt resistor, changing its resistance or turning it into an open circuit.
Blown Fuse. The fast-blow fuse disconnects the input, the speed depends on how severe the overcurrent is, from milliseconds to a few seconds - hopefully before damage has occurred. If the overcurrent is far above the tripping current, which is usually the case for a short circuit, it only takes milliseconds before fuse disconnection. But if the overcurrent is very close a fuse's tripping point, disconnection may take a few seconds (some multimeters have a 10 sec. max label on the current input terminal, which means it's safe to work near the the current range's upper limit, as long as you stop quickly before the fuse is blown).
For extra protection in case of overvoltage, the Fluke 87 has these designs: a diode bridge (I labeled it "OVP") would short-circuit the input to ground if the voltage across the shunt resistor is greater than the voltage drop of three silicon diodes (~2.1 V) for both positive and negative overvoltage. Furthermore, in case of overcurrent, the sensing line is isolated from the input current via a 100 kΩ series resistor (I labeled it "OCP"), ensuring that almost no current is fed directly into the data converter under a fault. In the milliamp range, the 110.1 kΩ resistor is also in series.
Conclusion: For just a mild overvoltage, it's extremely unlikely to damage the multimeter ASIC. It's possible to damage the shunt resistor, but it doesn't happen that often, since the fuse usually disconnects the input quickly enough. Thus, just replace the fuse and it's probably fine. If you're really concerned, compare the current measurement results of another known-good meter.
Even if you're very unlucky, the only thing that needs replacement is an inexpensive shunt resistor (and possibly a diode bridge in the Fluke 87). However, for a serious overvoltage, overheating and dielectric breakdown may cause additional component and circuit boards damage.
When measuring the same voltage source, the higher the range, the more difficult it is to damage it. The 10 A range is the most difficult range to damage, as it only has a 0.01 Ω shunt resistance and dissipates minimum heat. Many resistors can easily take an impulse of several hundred amps.