One can have any number of businesses operating under any number of business names. Once one properly separates the records for the relvant businesses to enable full and proper accounts to be filed with the relvant parties ( mainly Revenue and the banks if they require them for facilities granted) then there would not normally be any problem.
Revenue only take a 'dim' view of matters which are against the law. As rightly they should.
The CRO have no interest normally in Business NAmes once the requirements of the Registration of Business NAmes Act 1961 is complied with.