Sensory fatigue is not specific to odor. We become desensitized to our clothes during the day because of sensory fatigue in our sense of feel. Sensory fatigue is modified is every modality in the same way: something interferes with information going to the brain.
Olfactory fatigue is more obvious because olfaction is matter-based, while seeing and hearing are energy-based. With energy-based senses, when the energy stops all traces of the energy are gone. Matter-based senses (small and taste) do not just analyze energy coming from the target, they actually bring a particle of the target into the senses. This takes time, as the particles must be removed before a new particle can be sensed.
Most animals sniff. Sniffing cleanses the olfactory receptors and brings in a new cloud of particles. Rabbits have extra flaps in their noses so the olfactory receptors are not exposed prematurely. Humans will move an object they want to smell from one nostril to another while sniffing.
Olfaction is our most primitive sense. The olfactory receptors are actually brain cells that come down through the ethmoid bone of the skull. The olfactory receptors appear to be a yellow spot deep in the nasal cavity, but they are actually direct connections to the brain. Sight and sounds analyze energy patterns coming from a target. Smell analyzes the actual target. Even dogs know that something is real only if it has a smell.
When a particle attaches to an olfactory receptor it causes the receptor to fire a current toward nearby nerve cells (neurons). Signals are sent to the brain by chains of neurons, each of which may get messages from other neurons, The most efficient way to control the message is to interfere with the signals going to the brain. If an olfactory receptor is not sending a signal to the brain it might mean that there is no triggering particle in this olfactory receptor, or it might mean that some other neuron has inhibited it. In either case, as far as the brain is concerned, the offensive stimulus does not exist.