With the exception of the government approved lottery and a couple of horse racing venues, gambling of any kind is prohibited in Thailand.
Blatant gambling devices such as poker chips, roulette wheels, etc are banned in Thailand. Multi-function gaming items, such as playing cards, are not. And likewise general board games are legal as well.
The playing card rule is that it is illegal to sell playing cards in Thailand, it is not illegal to use them to play non-gambling games. It is one of those laws that makes you go "what the .... were they thinking" However, the police and the courts have very wide latitude when it comes to deciding if the games, cards, etc are/were being used for any activity where money changes hands. And being a tourist is not an excuse before the judge. Plenty of foreigners have been sent home with a less than desirable deportation stamp in their passports.
On the other side of the coin, the Thais love to gamble and play cards, same as folks from elsewhere in the world. Friendly card games happen throughout the country, out of sight of the local constabulary, but not really hidden from view.
As a tourist visiting or an expat living here, you need not worry about having a board game or three to enjoy at home or to keep the kids occupied. But engaging in any gambling games runs the risk of fines, jail time or deportation.