Visit Frank's Database at Jupiter Target in the shopping mall and check the progress requirement for the reward you want. Database progress unlocks collectibles, including badges and decorations. A newly discovered Style is useful progress, but it is not automatically the same as completing every reward condition associated with it.
Find Frank at Jupiter Target
Return to the mall and look for the Jupiter Target shop. Frank's Database lets you review Styles, weak enemies, elites, bosses and layers. Use those entries to check the abilities and discoveries recorded in your playthrough.
Check the reward condition
Read the requirement attached to the specific reward, complete that objective, then revisit Frank and follow the available in-game prompt.
A concrete example is the Demonic Bomber Style Skill Badge: its reward screen identifies the requirement as obtaining Demonic Bomber ten times. That is an example of a repeat-acquisition milestone, not a universal threshold for every badge or database entry.
Before another dive, distinguish discovering a form for the first time from repeating the action requested by an unlock. Look at the actual skill name when a core offers a new Style, since another core from the same monster type can carry a different active skill.
Keep collectible rewards separate from combat upgrades
Frank's Database can unlock statues, pins, badges and stickers as your progress grows. Earned statues are displayed in the mall's Museum. These collection unlocks are separate from purchasing Zombie Skills at Bessie Burger or installing equipment at Ink on the Brain.
If an expected reward is missing, check the individual requirement first. There is no verified single completion count or universal button sequence that applies to all of Frank's rewards.