1. Alan refused to let Jan pay for her movie ticket, even though the two were not on a date. Alan has a hang-up about what is socially acceptable and who should pay.
2. Please don’t use that kind of language or tell that kind of joke around me. I’m old-fashioned, and I guess I have a hang-up about such crude behavior, especially in mixed company.

The expression is used to describe an obsession or concern that may be regarded as somewhat eccentric, unreasonable or illogical.