Deep fry food in butter instead of vegetable oil

You can certainly deep-fry foods in clarified butter (also known as ghee) and in lard. In fact, there are many foods that are traditionally fried in these fats. They both have very high smoke points and are excellent for making crisp fried foods.

For example, Puri, Indian fried breads, are deep-fried in ghee (clarified butter). And many Southern USA and many Mexican deep-fried foods are meant to be fried in lard, such as hand pies or sopes. In fact, if you watch the videos of Cowboy Kent Rollins, you'll see that while many of his recipes say "frying oil", what he actually uses is lard.

As for the flavor question: yes, using ghee or lard will affect the flavor of what you're frying, but in subtle ways. Both of these fats are mild-flavored (at least, high-quality lard is). Generally, the extra flavor you get from the butter or animal fat is considered desireable; they fell out of fashion in the use due to concerns about cholesterol, not taste. Only foods that are meant to have a very light, airy batter (like tempura) are unsuitable for frying in animal fat.

There are some other animal fats that can be used for frying and deep-frying, such as beef tallow, schmaltz, horse fat, or duck fat. These have a much stronger flavor that is recommended for specific foods (for example, there are many aficionados of duck fat french fries), but aren't a good general substitute for vegetable oil.

One other caution: if you switch to frying with animal fats, you need to make special provisions to dispose of the used fat. It can't be safely poured down the drain. This is actually true of all deep-frying oil, but animal fats are a greater problem: they may clog your pipes as well as hurting the sewer system.

ADD: per @wjandrea below, clarified butter, ghee, and brown butter have different flavors based on the amount they were cooked while clarifying, which will affect the flavor of any fried foods made with them.