Hamsters and mice are not of the same species. Although they both belong to the order Rodentia, they are classified into different families in taxonomy and have remarkable differences.

The specific differences are as follows:
Taxonomic Differences
Hamsters belong to the family Cricetidae, while mice belong to the family Muridae. They are in different families in biological classification.
The ancestors of hamsters lived in the desert, whereas mice are widely distributed in the damp environments of cities and rural areas.
Physiological Characteristics
Hamsters have a round body shape and a short tail. Some breeds even have almost no tail at all. In contrast, mice have a slender body shape, with a long, thin, and hairless tail.
Hamsters have cheek pouches, which allow them to store food. Mice do not have this characteristic.
Living Habits
Hamsters prefer dry environments. They mainly feed on grains and are naturally clean. On the other hand, mice like dark and damp places. They have an omnivorous diet and are likely to carry germs.
Hamsters are usually active during the day, while mice are nocturnal animals.
Relationship with Humans
Hamsters are often kept as pets because of their cuteness and relative cleanliness. Mice, however, are regarded as one of the "Four Pests" because they spread diseases and damage the environment.
In conclusion, hamsters and mice have significant differences in terms of classification, physiological characteristics, living habits, and their relationship with humans. They are not of the same species.