The bigamy law of Indonesia reads:
(1) Met gevangenisstraf van ten hoogste vijf jaren wordt gestraft:
1°. hij die een huwelijk aangaat, wetende dat zijn bestaand huwelijk of zijne bestaande huwelijken daartegen een wettig beletsel opleveren.
2°. hij die een huwelijk aangaat, wetende dat het bestaande huwelijk of de bestaande huwelijken van de wederpartij voor deze daartegen een wettig beletsel opleveren.
This is different from the current Dutch criminal code
1 Met gevangenisstraf van ten hoogste vier jaren of geldboete van de vierde categorie wordt gestraft:
1°.hij die opzettelijk een dubbel huwelijk aangaat;
2°.hij die een huwelijk aangaat, wetende dat de wederpartij daardoor een dubbel huwelijk aangaat
The Indonesian code refers to an existing marriage being a legal impediment to a second marriage. The Dutch code refers to a 'double marriage'.
Under Indonesian law marriage is a religious act recorded by the state, never ever a civil act of union, and Article 1 of the Marriage Act of 1974 and related court decisions makes it extremely clear that a marriage is conducted by a religious leader, and the state is only recording the act that has already taken place. However, for Muslims, the state is involved in the religious proceedings, while for non-Muslims this takes place only after the fact.
Of course historically marriage in many countries didn't even involve a religious leader, let alone the state, and the Indonesian words for 'marriage', are both Arabic words meaning sexual intercourse and/or marriage, i.e. 'kawin' and 'nikah'.
The decision of the Constitutional Court in 46/PUU-VIII/2010 makes it clear that a purely Islamic marriage that does not involve the state's hand is still valid, providing it complied with Syariah law.
An unrecorded marriage in Indonesia is referred to as 'nikah siri'. This term is often used to describe any form of cohabitating couple who have not undergone any kind of religious ceremony whatsoever.
The term nikah siri is similar to that of نكاح العرفي or nikah 'urfi. https://en.wikipedia.org/wiki/Nikah_%27urfi However siri/sirri are used only in Indonesia and Saudi Arabia.
The practice of unrecorded marriages/nikah siri is very common in Indonesia, and gives rise to bigamy in that a man requires his wive(s) permission to marry further women.
According to all the sources I can find, if you perform (Muslim) nikah with the same person twice then that is meaningless, and while it's quite common to do so, the second 'marriage' cannot possibly be a marriage.
Since the second 'marriage' seems to be acceptable and indeed possibly desirable for familial harmony, while the second marriage does not create a new marriage, it does not seem that it is illegal to do so.
According to Catholic law, married people cannot remarry. They must have their existing marriage pronounced void. Article 1127 of Canon law provides 'It is forbidden to have another religious celebration of the same marriage'. 'Celebration' here refers to 'performing a marriage'.
It would appear that it might violate Indonesia's bigamy law for two Catholics to marry again.
Since it can be difficult to register a marriage many years later, it's likely common for Indonesians of all religions to 'marry again', ignoring the fact of the earlier religious wedding. It is not likely anyone would ever be prosecuted for this, since the state has the goal of regularizing everyone's paperwork, i.e. making sure people have marriage certificates and so on, and getting this done is more important than the specific precise details....