Why are some electronic devices rated for volts and some current?
The OP specifically asked about the Raspberry Pi, and why the current limits are not specified.  But actually, they are. And they should be for any microprocessor or microcontroller.  You just have to hunt for them sometimes.

Their are mainly two kinds of pins on the Raspberry Pi; power pins, 3.3V and 5V, and GPIO pins, which can be used as inputs or outputs.  All of the GPIO pins can also be re-purposed for various peripherals, such as UART, SPI, I²C, ADC and so forth.

In addition, all of the GPIO pins on a Raspberry Pi are rated for 3.3V only -- they are not 5V tolerant. 

Now to your question about current -- yes, all of the pins have a current rating.  That should be specified in the datasheet for the BCM2835, the processor found on the Raspberry Pi; however Broadcom has released only a partial datasheet which does not have all the required information.  Luckily someone else has made a Wiki titled "RPi Low-level peripherals" which has the information you are looking for.

For the power pins, the specifications are:

For the 3.3V power pins, the maximum current is 50 mA


For the 5V power pins, the limits are:

Model A: max current draw: 500 mA
Model B: max current draw: 300 mA


For the GPIO pins, the maximum current that the pin can either sink or source is configurable from 2 mA to 16 mA.

Sink means to draw current while the pin is at ground.  For example, you could have an LED with its anode connected to +3.3V via a resistor, and the cathode connected to a GPIO pin.  By making the pin an output, and setting it low, that will turn on the LED and sink current into the pin.  

Source means to provide current at the positive rail (3V).  You could have an LED with its anode connected to a GPIO pin via a resistor, and the cathode connected to ground. By making the pin an output, and setting it high, that will turn on the LED and source current out of the pin.

In a microprocessor's datasheet there is usually a figure for the maximum amount of current that all of the pins can draw together; i.e. you cannot expect to be able to sink or source the maximum amount of current (16 mA) from all of the 54 GPIO pins simultaneously.  But this number doesn't seem to be available as yet.