At the particular unit you mention (EGNR) the situation is thus:
The tower generally provides an aerodrome control service (callsign "tower") and a procedural approach control service (callsign "approach") on 124.5 and 123.35 respectively. This is known as "bandboxing"; i.e. one controller providing two services on different frequencies at the same time.
As soon as a radar service is required, the tower controller can no longer do both at the same time (it's against the law). A second controller (sitting at the radar screen behind the curtain at NR) will take over 123.35 and provide an approach radar service (callsign "radar").
Most larger airports (including Liverpool) are unable to bandbox tower and approach services as they provide 24hr radar. However, the callsign "approach" is frequently used by controllers when providing an approach radar service - they are theoretically providing both an approach control service (for example to non-squawking FIS traffic) and an approach radar service (for example to IFR traffic) on the same frequency at the same time. For the sake of convenience, most controllers will use one callsign or the other rather than switching between the two.
In practice (at PH, anyway) it seems to be the older controllers who call themselves "approach" on the r/t, and we youngsters who call ourselves "radar"!!
If you're not confused now, you never will be......