Do you have to acquire experience inside an EASA Part-145 Approved Maintenance Organisation?

Basically yes, but there are exemptions, for example the smaller “Part-M subpart F Maintenance Organisations” or inside the airforce. But in general the officials of the NAA (National Aviation Authorities) of an EU member state demand that the experience is acquired in a (aircraft) Part-145 approved maintenance organisation. You must show your experience in a logbook, the requirements for these logbooks can be Member State specific.