Things that people desire to touch in daily life are known to be limited to a number of specific targets (e.g., cats). The utilization of haptic displays to provide the experience of touching such desired targets is expected to enhance people's quality of life. However, it is currently unclear which haptic properties (e.g., hardness and weight) of desired targets should be rendered with haptic displays, and how they should be rendered. To address these issues, we conducted an experiment with 600 Japanese participants via crowdsourcing. Among the 600 participants, we identified potential users of haptic displays and analyzed their responses for each target. For each desired target, we identified the haptic properties in relation to which a "need for consistency" was felt by potential users between their expectations and actual impressions during touching. We also identified the haptic properties in relation to which a "biased impression" was held by potential users for each target. For example, potential users responded that cats were soft and that the actual impression of softness during touching needed to be consistent with their impression. Our results provide insights into the design of haptic displays for realizing desired touch experiences.