Small clarification about Indian law which might have been unclear from my previous message. Indian is also secular, though in a slightly different sense than in the European tradition. The relevant factor here is that in a society where the seat of effective government is far from the daily lives of most people, absent contrary legal principles, widespread social customs are grandfathered in to law. In other words, the government retains the right to legislate against it, and the fundamental rights like equality before law might automatically forbid certain practices, but in other matters, custom prevails.
As a different example, in India, marriages do not need to be registered, and births are often not registered. After marriage a woman does not have to formally declare her intension of using her husband's last name: she can legally continue to use her maiden (or otherwise previous) last name and start using her husband's last name or even both at the same time without a legal declaration. In some parts of India (like Maharashtra), the patriarchal structure is strong enough that at a wedding, the husband is allowed to ritually rename his wife. I suspect the law would recognize such renaming as well, though in the society I am familiar with, I have not met a husband who would dare to exercise this "right", so I do not know.
In India, it is also legal to change one's last name, or drop it altogether, by a simple affirmation in court and public notice. I suspect it would be legal to add multiple last names, though I know of no such case.