In Japan, family names are generally carved on tombstones like 田中家之墓/Tanaka family's grave. 田中/Tanaka is a one of very popular family name. 家/ke in this context means family. 墓/haka means grave. Since the rate of cremation in Japan is over 99%, your ashes are placed in a tomb. In addition, a few family members' remains are usually placed in a single tomb. The full names of all the family members are carved on the side of the tombstone or a small stone placed beside it. In other words, your ashes are supposed to be placed in the family grave to which you belong. If a woman marries a man and takes his family name, she will belong to her husband's family. If she unfortunately dies young before her husband does, and if he is conservative, her remains will be placed in his family grave where the ashes of his parents or grandparents are placed. Needless to say, you can have your own grave if you want and are allowed to do so.
Recently, individuals are emphasized more. More and more people don't stick to the traditional idea of which family you belong to. However, at wedding parties, the expression "○○家 と○○家の結婚披露宴/ a wedding party for ○○ family and ○○ family" is always used. Other than that, you can sometimes find the tradition in daily life.