In this context "firmware" refers to code essential for the running of the device that is loaded by the device's driver when it's initialised. It isn't actually stored permanently on the device.
This is often referred to as "microcode" (cleverly abbreviated to "µcode") instead of firmware because it's small enough that it can be loaded onto the device each time it's powered up.
The "firmware" or microcode you load on one OS shouldn't affect the operation of that device on another because booting the other OS will load microcode from its own driver.
Obviously there are some devices which have a permanently flashed "firmware" on it, like complex self-sufficient devices such as routers, smartphones, NAS devices etc, but in those cases the firmware is pre-loaded onto the device already, rather than loaded on at boot via a device driver in the OS. Such firmware can be "updated" by obtaining a special firmware update and installing it in the way recommended by the manufacturer but it then replaces the permanent firmware that was already on the device.
A Wifi adapter, such as a PCI/PCIe/USB based one, is almost certainly not storing any permanent configuration settings or firmware on the actual device and all code will be loaded by a driver on the PC each time the device is initialized.
During boot up, a message briefly appears saying that firmware files b43/ucode5.fw and b43-open/ucode5.fw not found. Go to http://wireless.kernel.org/en/users/drivers/b43#devicefirmware and download the correct firmware for this driver version, carefully read all instructions on the website.
This might not actually be the best way to install the firmware. This is a general Linux-based message that isn't tailored specifically to Ubuntu. Ubuntu actually provides some easy-to-install firmware packages for common devices that you can simply install with apt-get, and may be a more robust alternative to downloading the raw firmware files from upstream.
In this case you may be able to install the b43 firmware with the Ubuntu command:
sudo apt-get install firmware-b43-installer
Should this command not work for you there is some more information about that model of wifi adapter in this question.
The reason it's not installed by default but needs to be installed this way is almost certainly due to licensing restrictions; the microcode part of the driver is not open source and probably has a license that restricts it being distributed with Ubuntu - but the user (you) are allowed to download and install it yourself.
