Yes, sort of. Illiterate people could "sign" using hand prints, which is a reasonably reliable biometric (totally anecdotal, but my university's experience was <10% false identification) that's a bit easier to authenticate by the naked eye. Prints of the finger (more than just the tip) could also be authenticated based on feature such as lengths between joints.
(Marriage contract from the early Republican Period.)
The name these movies usually give for this practice,
huāyā (花押/畫押), is a much broader subject. That tradition developed sometime between the Jin and Tang Dynasties, when paper became popularised as a writing medium. In many regards it is functionally similar to western signatures, and could be seen as a less formal alternative to using seals. It was a convenient way of marking documents for all sorts of people, and not restricted to the illiterate by any means.
In its earliest forms, the practice was essentially writing one's name in a slightly fancier way. Over time, people found increasingly creative ways to make their marks unique. By the Song Dynasty, some huāyā had become wholly unrelated to the signer's actual name. Others don't evenresemble real characters anymore.
((a): huāyā of calligraphist Wang Xizhi
王羲之. (b): Emperor Huizhong of Song
趙佶. | (c): Emperor Sizhong of Ming
As a practical matter, keep in mind how writing had to be made historically. The traditional implement for proper writing in China, the ink brush, is actually very hard to use. Even if you know how your name should be written, an average farmer might not have the practiced dexterity necessary to actually draw the character legibly. Thus, for many people drawing such signatures with fingers would have been a logical and obvious improvisation.