A school does not necessarily meet the definition of a public authority [1]; I would say it's not; it depends on individual EU countries to legislate on that definition.
Furthermore, in Ireland, for example, fines cannot be levied against public authorities: https://www.irishtimes.com/news/crime-and-law/public-bodies-...
[1] https://www.lexology.com/library/detail.aspx?g=e71ac55f-566b...