Why do Chinese and Hindi have more terms for relatives than English does?

Kinship terms tend to reflect past social structures. Thus Chinese maintains a sharp distinction between relatives on the mother's side (cognates) and relatives on the father's side (agnates) because the religious and legal concept of a family was traditionally based on male descent. (A wife formally left her father's family and became part of her husband's, even to the point of sacrificing to her husband's ancestors.) Therefore it was important to distinguish agnates from cognates.

Traditional Confucian teaching was also very hierarchical. A wife was subordinate to her husband, a sister was subordinate to her brothers, and elder brothers held authority over younger brothers. Therefore there are separate terms for older and younger brothers, older and younger male cousins, etc.

Such features have nothing to do with cultures being more or less "family-oriented", but may reveal something about social structure. Numerous societies distinguish relatives based on whether the person is on your maternal versus paternal line. For example, in Logoori, the word translated as "uncle", koozá refers to the male siblings of your mother but not the male siblings of your father. The word translated as "aunt", séénge, refers to the female siblings of your father but not the female siblings of your mother. You refer to your paternal uncles as "father" and your maternal aunts as "mother", hence "my fathers" and "my mothers" is not as marked as it is in English. However, this distinction stops with your parents, so there is no difference between maternal and paternal grandfathers, or grandmothers. The distinction between matrilineal and patrilineal societies is important, but not all-determining, in studying kinship systems. The study of kinship systems is a well-studied area of anthropology and linguistics.