Half-Identical Twins: Subverted, these two aren't siblings by blood and certainly aren't the same age. However, this is a popular way to see them portrayed in fanon due to them having the same haircut and therefore looking incredibly alike.
Lady and Knight: A popular fanon rendition, though Switzerland is more Trigger Happy than most White Knights and Liechtenstein is yet to be seen in serious danger after being taken under his wing.
Not Blood Siblings: Of course, this also means that there's nothing really standing in the way of the obvious Ship Tease and tension between the two.
There's also the subtext of their relationship being so close as to be borderline incestuous.
And, of course, since they're not actually related, live in the same house, are excessively intimate with each other and didn't grow up together either, a possible latter separation would be devastating to both of them, so the most common speculation is that they would end up like Austria and Hungary one day.
Rags to Riches — Both of them. On one hand Liech was this close to dying when her lands went bankrupt due to World War I and the Great Depression, until Swiss took her in. On the other hand, Switzerland's own economic situation was very precarious when he found and adopted Liechtenstein, for exactly the same reason. Now both countries are extremely rich, with Liechstenstein as one of the richest nations in the whole world.
Ship Tease — Pairing the two together isn't that much of a stretch, given how they act to one another.
Switzerland: Man up or I'll beat you with my Peace Prize!
IcyGreen Eyes: For his grumpy, impersonal nature to anyone not his sister. His eyes can also be an allusion to the diverse landscape of icy mountains, clear blue lakes and rivers, and the green fields and forests his land is often noted for.
Ineffectual Loner: He's shown to be rather reclusive and doesn't seem to like parties, not to mention his infamy of keeping others out of his land by shooting at them. But at the same time, he still ends up talking to Austria in modern day and when France comes to visit him, he openly complains and seems very annoyed but doesn't actually make an effort to throw him out. Then there's Liechtenstein...
Fan Nickname: She doesn't have an official human name, but some fans call her "Lili Zwingli" to bypass that. It comes from a shortening of her nation name and from the Lily flower, which is sometimes considered to be a national flower of Liechtenstein.
Himaruya himself in a recent blog post suggested her likely human name to be Erica/Erika, Sisia, Elise, or Eva, with her possible surname Vogel.
Proper Lady: Has the personality traits: hardworking, mature, polite, very competent in housework, sweet and devoted to those she loves. In fact, she is so far the only girl to be considered as such(apart from Ukraine). Just compare with either Hungary, Belarus or Vietnam
Rags to Riches: Her living standards were still kinda shaky and unwell until World War II was fully over, even with Switzerland's help. And now she and her lands are freaking loaded.
Sexy Santa Dress: Drawn wearing one during one of the Christmas Events. Albeit not incredibly revealing, it was a lot shorter than pretty much all of her other dresses.
Tertiary Sexual Characteristics: Due to her, um, lack of secondary characteristics, Switzerland just buys her a blue hair ribbon so no one mistakes her for a boy again.
The Gunslinger: In Fanon as a consequence of her emulating Switzerland constantly. She has been occasionally shown in some sort of military training with him, carrying a rifle on her back, so somewhat confirmed.
The Unpronounceable: At one point in the dub, the Narrator apparently has trouble pronouncing Liechtenstein's name correctly. When she does, the child audience cheers.
What Could Have Been: At first Himaruya considered having her emulate her brother even more, with a harsher personality and a cold manner of speech.