Flying robots, such as quadrotor drones, offer new possibilities for human-robot interaction but often pose safety risks due to fast-spinning propellers, rigid structures, and noise. In contrast, lighter-than-air flapping-wing robots, inspired by animal movement, offer a soft, quiet, and touch-safe alternative. Building on these advantages, we present \textit{Cuddle-Fish}, a soft, flapping-wing floating robot designed for safe, close-proximity interactions in indoor spaces. Through a user study with 24 participants, we explored their perceptions of the robot and experiences during a series of co-located demonstrations in which the robot moved near them. Results showed that participants felt safe, willingly engaged in touch-based interactions with the robot, and exhibited spontaneous affective behaviours, such as patting, stroking, hugging, and cheek-touching, without external prompting. They also reported positive emotional responses towards the robot. These findings suggest that the soft floating robot with flapping wings can serve as a novel and socially acceptable alternative to traditional rigid flying robots, opening new possibilities for companionship, play, and interactive experiences in everyday indoor environments.