I'm going to describe a couple of cartoon faces for human and anthropomorphic characters.
Black Dot Face
Character has black dot pupirises. These change into other graphic shapes to represent shifted eyes, or they may appear within outlined eyes. The thin eyebrow lines may also indicate the eyes or the brow due to the lack of actual mass.
Early Mickey Mouse, like many Disney characters has an especially tricky one. He has dot eyes, outlined eyes, and a blackened hair-like brow that suggests eyes by creating negative space. When eyes are used, they may also include optional eyelids. Sometimes there's even an outline close to the brow on the upper part. This joker outline allows for the perceptions of the eyes to be widened without drawing the full eyes, taking up too much room on the face. Additional features may be added for effect, like eyelashes or defined lips.
This fades out as the eyes become set. The fleshy cartoon skin obsoletes the purpose of shifty graphic brows. Once the character's looks become set, they become more specific, concrete things rather than amalgamations of shifting parts. They become locked in more and more to specific shapes. Then we see solid forms with fluidity.
This clever face gets replaced by the noodlebean cartoon face. There's fully defined features, with some cartoon obsoletions. The eyes are set, the cartoon skin is loose in order to put it into the right shape. The brow(generalizing here) is exaggerated, a fleshy thing, and it has taken the place of eyebrows and graphic brows. Eyebrows are now exaggerated merely as eyebrows and eyes as eyes. The new forehead brow will still make advancements over graphicized eyes for emotional effect.
The only other formula I can think of is in anime. There's a generic set of facial features which transcend a lot of designs. I'll focus on one variant that seems especially popular at Ghibli.
The lipped mouth and nose are only fully shown in side views. They're otherwise graphicized, the mouth as a line and the nose only partially shown in line. A thick line on the top of the eye suggests the top of the eye cavity, and the pupirises/pupils-in-irises are vertically stretched into ovals. The graphicized features again allow for additions to be made while keeping the characters relatable by not showing certain less appealing, commonly shared traits. Features are sometimes drawn especially vaguely in further views.
These all take on the idea of adding facial details from a standard template that has less detail than a real person's face. Perhaps soon this will flip towards starting with all the normal features and then removing them for effect. This would be much easier to do with computer animation. In animation, there's always the possibility of getting rid of people in favor of transhumanoids who look different. Those poor suckers in live action might have to put up with the boredom of real people, but that's not true in animation. I'm rather perplexed as to why most animated films aren't set on other planets with creatures that have their own unique systems of communication or in the fourth dimension. When you sit down and think about it, they would be if animation explored its possibilities.