This meme seems to be a conflation of two very different things.
First, Hyakuya Akane from Owari no Seraph (the girl on the lower part of the image) does not share the low below-the-shoulder ponytail that the women in the top of the image share; she has a loose braid. Today I was handed this event flyer by a friend (a Japanese college student involved in subculture), which features a girl with a one-sided, front-of-the-shoulder loose braid and long bangs. I mentioned this SE question to her and she said the hairstyle has no association other than looking gentle and seemed incredulous that someone would come up with such an idea.
A loose braid, or two, or a single low ponytail on one side at the neck, or two low/loose pigtails, or are fairly common for contemporary pre-teen/teen girl character designs, since this is a current fashionable style among real-life Japanese pre-teens through college students. So Akane doesn't necessarily fall within a grouping of mother figures or people fated to die any more than the following characters do:
Second, mothers can be given the low ponytail because it is one of a number of soft, gentle hairstyles (contrast this to styles such as a short page-boy clipped cut, or a long past hip-length luxurious cut, or spiky hair). Not all mother characters, but many are portrayed with a nurturing, warm personality. However, plenty of kind mother characters in anime -- both those who die and those who don't -- sport other hairstyles. (Also, the hairstyles in the pictures in the meme are not that strongly similar to one another when you look at the styles of the bangs.)
Third, there are largely no moms in classic literature (a wealth of orphan stories: Heidi, Anna of Green Gables, Tom & Huck, David Copperfield and basically all of the Dickens' boys, little princess Sara, Mary of the secret garden, etc.) or in Disney animation either. Dads trying to marry off their daughters, yes. Cute, plump, bumbling dads, yes. Absentee fathers, yes. But in many cases, we have orphan protagonists with no parents around at all. We can feel empathy for the characters whose loved one died and a death can be a strong motivation for characters to resolve to take a stand against evil, but more importantly for drama, for suspense, for excitement: without a mom to take care of them and worry about them and discipline them and tell them which things are too dangerous to do, orphans are freed up to go on adventures. No questions asked! No curfew! If you have an eccentric, sweet little papa or dad who works long hours or lives separately from you, it's also easy to slip out of the house to go on adventures. Where are the moms? Well, evil stepmoms are okay, since the caring mom is already dead and safely out of the way for the story to move forward. A common plot element in stories is that kids only get an adventure if mom is already dead. So the unfortunate message we get is, motherhood is boring, and mothers make their kids' lives boring. Mothers who exist at the start of a story, irrespective of hairstyle, will often need to be killed off in order for the protagonist to achieve the freedom to do dangerous things that a responsible mom wouldn't stand for.