Can a travel agency really file an absconding case over my visit visa?

Your instinct is largely right — absconding cases in the employment sense are filed by employers against their sponsored workers through MOHRE, and a travel agency that arranged a visit visa has no such power over a tourist. What a visa sponsor or host can legitimately do is report a genuine overstay, meaning a visitor who remains in the country after the visa expires. If you leave on or before your exit date, there is nothing real for them to report, and a threat to file something anyway is empty. What the agency is doing, however, is not empty in another sense: demanding extra money under threat of an official report is extortion, and that is a matter for the police, not something to negotiate. Keep every message, voice note and payment demand. Do not pay. Report the threats to the police in the emirate where the agency operates, and consider reporting the agency's conduct to the authority that licenses it — travel agencies have regulators, and shaking down visa clients is licence-threatening behaviour. Then make sure your own side is clean: exit before the visa expires and keep proof of departure. If the agency escalates or you discover anything was actually filed against you, a lawyer can deal with it and advise on reporting the agency formally.