To be honest this is another area where the standard is a little bit fuddled (mixed up). Just for simplicity I'll refer to the the two parts of the system as the "adaptor" and "unit", where the unit gets a 12V supply from the adaptor.
To reconfirm, the origin of Class I, Class II is for mains supply (i.e. 100Vac, 230Vac, 415V etc). In the normal course of events, the adaptor would be seen as a completely separate device, full tested to meet standards. This means the unit does not need to worry about mains related issues at all. Historically the unit would have been designated Class III. This classification actually existed in the 1977 edition of IEC 60601-1 (which I am sadly old enough to have actually seen ...).
The 1988 edition got worried about the adaptor not having enough insulation or too much leakage for the medical environment, and as such the definition of Class III was ripped out. In effect, the standard applied to the system of the "adaptor + unit" as a single device. However, in principle, all mains related requirements (including Class II marking) were only required for the adaptor, not the unit.
The 2005 edition relaxed this somewhat as for many situations, non-medical adaptors are fine, or can be handled as a system under Clause 16.
But they did not re-introduce Class III. So, currently Clause 6.2 reads as if the unit is Class II.
As with the 1988 edition, if you search through the standard for all the references to Class II, you will find they are all mains supply related and hence not applicable to the unit. As such, the classification has no practical impact, and only serves as a point of confusion.
One common mistake is to mark the Class II "box in a box symbol" on the unit, even though it has no mains parts. In the 1988 edition the exclusion was hidden in "Table II", which meant it was easily overlooked, hence it was a very popular mistake. In the 2005 edition, the exclusion is less hidden in the title of the clause - 7.2.6 - "Connection to the supply mains", where "supply mains" is a defined term which essentially means the wall supply; but again easy to overlook if you miss the title or definition, and especially with the history of mistakes coming from the 1988 edition.
In fact, it should be noted that several items like supply voltage, current or power, dc symbol are not required to be marked on the unit, nor is the power input test applicable to the unit. These are all only required for equipment which connects to a mains supply.