What do you call a sibling in law?

Do your siblings in law refer to you as "Anee-san" or older sister if you marry their older sibling or do they refer to you as an imouto?

I know in Vietnamese culture, if you marry the older son, you are considered the older sister even if you are younger than the other siblings. Is this the same in Japanese culture or different?

Do you mean like when you marry a man who has younger brother or sister?

It can depend on the family or the age difference, but traditionally you will be called onēsan (O’neesan), おねえさん (お姉さん), meaning big sister.

In Japanese, the "in-law" part of siblings in law is "giri no," so the four possibilities would be:
giri no oni-san = older brother-in-law
giri no ototo = younger brother-in-law
giri no one-san = older sister-in-law
giri no imoto = younger sister-in-law

Whether your siblings in law use the "older" or "younger" version depends on who you married, not our actual age. For example, my wife has a younger brother, who is actually a year older than me. However, because I married his older sister, I'm technically his "giri no oni-san"/older brother-in-law, even though I'm actually younger than he is.

So yes, like in Vietnamese culture, if you married a man, you'd become "giri no one-san" to his younger siblings.

However, there are a few important things to remember here:

1.
Even if you're technically "giri no one-san," your siblings in law will never call you "one-san" or "giri no one-san" when they're speaking directly to you. Instead, they'll just call you by your name plus -san. It would be extremely weird if my wife's brother called me "oni-san."

2.
Your siblings-in-law probably won't call refer to you as "giri no onse-san" even when talking about you to other people either. It's far more likely that they'll refer to you as "oni-san no oku-san," or "my bother's wife."

3.
Finally, being "giri no one-san" doesn't give you the same sort of age-based deference that siblings might show to their actual by-blood older siblings. Japanese families aren't hierarchical in that way. My brother in law is always nice and polite when we're talking, but there's never an atmosphere that because I'm his "giri no oni-san" he has to constantly make a show of his respect for me or seek my approval for his actions.

I would say it is the same as in Vietnam as you describe it.
And by the way, my aunt (my mother’s younger brother’s wife) calls my mother “o-ne-san,” never by my mother’s name. When my aunt refers to other relatives about my mother, I have heard “(name)-ne-san,” to distinguish which sister-in-law she is referring to.
Maybe it differs by generation. Too bad I don’t have any brother or sister, and my spouse is non-Japanese, so I can’t tell you what people in my age bracket would do :)

I understand what you mean.

Suppose you are 25 years old. You have a brother who is 27 years old. The brother marries a 24 year-old woman named Hanako. Are you expected to call Hanako "o-nee-san (big sis)"? The answer is "yes".

But in real life, that doesn't happen very often. Everybody sort of ends up calling that person "Hanako-san", even siblings-in-law who are several years younger than her. In fact, it really feels weird to be called "o-nee-san" by a sibling-in-law who is older than you but younger than your spouse. At least I hated it when that happened to me, and I know I am not alone, because I saw a similar situation portrayed in a fiction TV drama.

But technically speaking, Hanako is indeed your "giri-no-ane" instead of your "giri-no-imouto".