There are times when Wind Chill Advisories or Warnings are issued, but the wind chill effect is minimal due to no wind, but air temperatures dropping to the Advisory or Warning thresholds.
To understand why, it's important to understand the difference between temperature and wind chill. Temperature is a measure of how much energy molecules/particles have. They have more energy when heat is added and less when it's cooler, so it's a good measure of how hot or cold something feels. This is what inanimate objects experience.
For example, water won't freeze until the air temperature drops below freezing, not the wind chill.
Wind chill is really how we as humans feel. It is a measure of how fast our body loses heat. Wind will actually accelerate our body's heat loss because it pulls heat away from our skin faster than if there wasn't wind. Therefore, wind chill is the temperature that we'd lose heat at the same rate as if there wasn't any wind.
Therefore, the Wind Chill Advisory and Warning are issued based on the impact- which is how fast frostbite can set in. The human body doesn't care if the rate of heat loss is due to a really cold temperature or wind chill, it's just what's happening and both can cause it.
Pets do feel wind chill, but likely isn't quite the same calculation as it is for humans. Still, when it's too cold for you to be out without covering up it's likely too cold for house pets to be out without extra layers.