This is not correct. The cultural tradition for most of Chinese history was that the woman did not take her husband's surname, and this is mostly the same today except for some exceptions in Hong Kong or overseas Chinese communities. In fact, Hong Kong is one of the few places where Chinese women can take their husbands' surnames, but this is not the necessary rule. For example, in Hong Kong the politician Carrie Lam, or Lam Cheng Yuet-ngor (林鄭月娥), was born with the surname Cheng (鄭) but married a person with the surname Lam (林). It's similar with the politician Regina Ip, or Ip Lau Suk-yee (葉劉淑儀), born with Lau (劉) but married to an Ip (葉).

Let's have a hypothetical woman, He Ying (何英). Last name is He, first name is Ying. Let's say she marries a man, Gao Qiang (高強). She might then become known as Gao Taitai (高太太) or Gao Furen (夫人) in some Chinese communities, which will mean something like "Mrs. Gao." But those terms are titles and the woman in question didn't change her name. Gao Taitai and Gao Furen would be better translated as "Wife of Gao." So this hypothetical lady should be thought of as "He Ying, Gao's wife" and not "Gao Ying, Gao's wife." She's still He Ying (何英), not Gao Ying (高英). And this thing about Taitai/Furen is widespread in only certain Chinese communities. In Mainland China a married woman would probably just go by her last name.

Here's a timeline for Taiwan.

1. Pre 1948: during the Japanese era, Taiwan civil disputes were resolved with consideration to Taiwanese customs, as per order of the then Governor-General.
2. From 1948 to 1985: entering the Chinese (ROC) era, and Chinese civil law applied. It was written down in law that married women shall affix on their surname that of their husband, unless otherwise agreed upon.
3. From 1985 to 1990: efforst to amend the Civil Law stared in 1975 in order to catch up to United Nations human rights standards. It took 10 years for this amendment to be finalized, but most changes were just play on words that tried to preserve the old ways. The article on woman taking on the husband's surname did not change. The committee responsible for these amendments had only 1 woman out of 13 members. From meeting records it was clear that the rest of the committee paid no thought to her opinions. The parliament (Legislative Yuan) also had only 1 female member. As a result, the amendments pretty much had no effect in the real world.
4. From 1990 to 1995: women's rights social movements. After holding many hearings and public petitions, they moved to have the Grand Justices do a constitutional interpretation on women's rights, successfully having the parts in civil law discriminating women be declared unconstitutional. The Grand Justices gave a 2 year grace period for the parliament to make appropriate amendments. In the end it took 4 years.
5. In 1995: new civil law. The article on women's surname was removed. In its place, the law now states that women are free to choose whether to take on the husband's surname, and are free to restore their own surname afterwards. (PS: my mother did exactly this, had her surname restored.)

In general, the change in Taiwan was simply a part of the wider women's rights movement, which in turn was a part of the even bigger gender equality movement.

TL;DR: before 1995, women need to be prepared to piss off their in-laws to keep their surname. After 1995, the law became neutral and no longer default to women taking on their husband's surname.