While the United States holiday was adopted by some other countries, existing celebrations, held on different dates, honoring motherhood have become described as “Mother’s Day”, such as Mothering Sunday in the United Kingdom or, in Greece, the Eastern Orthodox celebration of the presentation of Jesus Christ to the temple (2 February of Julian Calendar). Both the secular and religious Mother Day are present in Greece. Mothering Sunday is often referred to as “Mother’s Day” even though it is an unrelated celebration.

In some countries, the date adopted is one significant to the majority religion, such as Virgin Mary Day in Catholic countries. Other countries selected a date with historical significance. For example, Bolivia’s Mother’s Day is a fixed date, commemorating a battle in which women participated to defend their children.

Some countries, such as Russia, celebrated International Women’s Day instead of Mother’s Day or simply celebrate both holidays, which is the custom in Ukraine. Kyrgyzstan has recently introduced Mother’s Day, but “year on year International Women’s Day is certainly increasing in status”. Source via Wikipedia

TL;DR In many countries the modern form of the holiday is strongly influenced by the US version but may be centered around an pre-existing holiday intended to celebrate motherhood from their unique cultural/religious traditions.