You are mistaken on how the Arduino wire library works.
The Arduino wire library makes no assumption as to what is on the other end of the i2c connection.
It merely implements the i2c protocol.
That is why it works with all kinds of devices like i/o expanders, thermometers, LEDs, etc....
That i2c backpack uses a MCP23008 which is not used very often.
Nearly all the backpacks out there use the PCF8574.
But even when a library supports the i2c chip used,
it doesn't mean the library can make the LCD work.
This is because the library must program the output pins of the output port
of the i2c chip to talk to the LCD based on how the i2c chip is wired to the LCD.
Not all boards wire the i2c chip up to the LCD the same way.
There are libraries out there that support the mcp23008 chip,
(the maker of that board/lcd even supply one),
but the only other board I've ever seen us it is the adafruit backpack.
Are you using the library that they supplied?
--- bill