Can the employer do this, or is it automatically canceled somehow?
No, not necessarily.
Registration is automatically terminated if:
person's employment,
reinstatement based on a court decision that has entered into force,
a two-time refusal to offer a suitable job, and for an unemployed person who is looking for a job for the first time and does not have a profession (specialty),
two refusals to offer professional training,
failure to visit a territorial body of a central executive body that implements state policy in the field of population employment and labor migration without valid reasons within 30 working days from the day following the day on which it made a decision on such a visit,
conscription for term military or alternative (non-military) service.
Article 45 of the Law of Ukraine "On Employment of the Population" defines many other grounds for canceling the status of unemployed - https://zakon.rada.gov.ua/laws/show/5067-17/conv#n433