Pew Research has survey data germane to this question. As it stands, a clear majority (79%) of opposite-sex married women changed their family/last name to their husband's.

But for never-married women, only a third (33%) said they would change their name to their spouse's family name. 24% of never-married women were unsure whether they would or wouldn't change their name upon marriage.

From this data, I would conclude that while the trend of taking the husband's last name is fairly entrenched right now, the public's attitude are changing and we might expect the popularity of this to diminish over time. The detailed breakdown by demographic shows that the practice was less common (73%) in the 18-49 age group than in the 50+ age group (85%).

However, some caveats: the survey questions did not inquire into whether the never-married women intended on ever getting married; it simply asked "if you were to get married...". So if marriage as a form of cohabitation becomes less popular in the future, then the change-your-family-name trend could be in sharper decline than this data would suggest.

Alternatively, the data could reflect differences between married and never-married women. Perhaps never-married women -- by virtue of not being married yet -- answered "would not change name" because they did not yet know what their future spouse's name is. No option for "it depends on his name" was offered by the survey. Never-married women may also more-strongly consider the paperwork burden -- USA specific -- for changing one's name.

So does this help answer your question? Eh, only somewhat. Younger age and left-leaning seem to be factors, but that's a far cry from cause-and-effect. Given how gradual the trend is changing, it's more likely that the practice is mostly cultural. If so, then the answer to "why is cultural practice XYZ a thing?" is always "because it is".