But what is the purpose or reason for saying trffic believed to be.....I mean isn't it clear that he is who you think who he is by the tranponder code? We never identify those traffic, we just shout the callsign and, for an example, tell them you got trffic to your right or 2 o'clock at 10 miles or you are number two in the sequence behind someone. Do yo mean this cannot be done unless I hold a radar rating? In our aerodrome all the VFR traffic have pre-assigned transponder codes and they always use the same code. When they are airborne we mannually input the callsign (which is also the rgistration of the ACFT) corresponding to the code.
The reason for saying traffic believed to be you is that it enables a radar controller to offer information without positively identifying the aircraft or committing to providing a radar service (in some countries and some airspace classes this is an option). Aircraft are not always easily identifiable - sometimes the rules allow the use of conspicuity or common use codes which mean that more than one aircraft will have the same squawk.
Providing traffic information in the way you describe seems OK (to me anyway) without a radar rating. It's effectively providing a FIS with the aid of radar which is something the ICAO rules permit and is done in many parts of the world.