No, it does not.
At least it can not in any useful way. If you attempt this, it can only drive one of the 8 pins at a time. So in some manner, it can allow you to access 8 output by using it in reverse. But each is only connected temporarily, so it just isn't very useful if you attempt to use it that way.
Another problem with trying to use an analog mux chip for output is the on resistance. When you make a connection through the 74HC4051, the switches act like resistors. Running at 5 volts, the resistance is about 70 to 100 ohms. Running at 3 volts, it's higher (the chip specs don't say how much worse... which probably means it's quite a lot worse). When using the chip for inputs, the resistance is not a problem. But if you try to use it for output, the resistance also limits its usefulness... as if only being able to temporarily drive one of the 8 outputs at a time wasn't a big enough problem!
Usually for outputs, a shift register is used. The 74HC595 is the most popular type.
For driving relays, which require more power than most normal logic chips can provide, you should consider a part like TPIC6B595. Digikey has it as part number 296-1956-5-ND.