within the SEP(L) class with only familiarisation training (e.g. reading the POH) required between types.
I very much doubt reading POH would be enough as a 'familiarization' training. You will need endorsement for a type (in other words flight instruction in the type) and often other additional practical obstacles would stand in the way (like flying club or insurance requirements). If it was so easy to move within SEP(L) someone could fly 172 and then after 'familiarization' jump into say Piper Mirage or Cessna 210. Anyway, this is how it looks from the FAA point of view and I imagine things are similar in Europe. By the way, no SEP under FAA, it is SEL (single engine land).