Thinly disguised (usually hyphenated) is used to describe something that is officially/formally said to be one thing, but that most reasonable people would recognise as actually being something else.
In this specific context, the notable items are the expensive gifts. They are officially gifts (i.e. optional items given to the bride's family, in celebration of the occasion); but the article implies that they are actually expected to be given, as a payment for the bride.
The value of the gifts is the price that has been paid to the bride's family, by the groom's family, in exchange for the bride's lifetime commitment of marital servitude.
The expectation that the bride's family should receive a payment is an unwritten/unspoken rule, and referring to these payments as gifts lends a veneer of traditional respectability to the process of buying another human being.
You may also see the term thinly-veiled, which might be a little too on-the-nose for an article about weddings. Something that is thinly-disguised may also said to be something, in all but name (e.g. these "gifts" are a payment, in all but name).
A similar practice of exchanging goods or money in return for marrying into a family is referred to as a dowry, but that is more-specifically where a bride (or bride's family) gives money to a groom, in payment for taking on caring for the bride.