The short answer to your question is, no, you don't have to separate your "VO2Max power" between flat rides and climbs. However, specificity matters and the pedal force/pedal speed plots show that the demands of flat rides and steep climbs differ, so you may want to keep that, plus your recovery capacity in mind when you train.
The longer answer requires some explanation of VO2max, your power "at VO2Max", the differences between 5-minute efforts and longer efforts, and power application within the context of a race.
VO2Max is the maximal amount of oxygen you can consume. It is usually normalized by body mass so is measured in ml/min/kg. It is not independent of the sport you are doing so VO2Max can differ between cycling, running, rowing, or xc skiing, but the dependence among trained athletes is relatively small, perhaps 5% or so, usually less than 10%. The difference can be greater among athletes who are novices to a particularly sport, but here we are talking only about cycling, and about riding on the flat vs. climbing up steep slopes.
A larger issue is that although we typically say that the oxygen you consume during maximal effort over 5 to 8 minutes is a good predictor of your VO2Max, it is often hard to spot a truly maximal effort from data collected during a race, especially a mass start race. The reason is that during a mass start race, tactics will depend on context, so picking out the "peak 5-minute" effort may often be an interval where you still have a long way to go to the finish line instead of collapsed in the gutter on the side of the road. For hill climbs, especially but not limited to TT hill climbs, once the climb is over you often (though not always) have a respite and can recover a bit on the descent. Therefore, just looking at 5-minute peak power during races typically won't be a good estimator of your true VO2Max. That is, your "VO2Max power" may not differ much between types of races but your 5-minute peak power can be quite a bit different, depending on the race.
That said, a look at the pedal force/pedal speed plots shows that hill climbs can require a different pattern of cadence and force than rides on the flat; and some riders may have an easier (or harder) time generating those patterns on particular types of terrain. Here is a plot for one rider showing his overall cadence/crank torque (very similar to a pedal force/pedal speed plot) during hill intervals. The upper left panel shows his overall pattern, the upper right panel shows that the elevation profile for the ride; as can be seen, he rode out to a nearby hill, did four climbs and descents, and returned home. The climbing portion of the ride is shown in red while the flatter parts are shown in black. The lower two panels show the cadence-torque patterns used for the climb and for the flat. It is evident that they are different.
Below, you can see cadence-torque plots for the same rider in three different types of races: a hilly (but not mountainous) road race, a flat urban loop criterium race, and a rolling time trial.
Largely speaking, cycling speed on climbs is mostly determined by power-to-weight, while speed on the flats is mostly determined by power-to-aerodynamic drag area. This means that power scales with speed differently on climbs vs. on the flats. Another thing that varies with climbs vs. flats is crank inertial load, and empirical research suggests that some riders are more "sensitive" to changes in crank inertial load than others, and will choose different combinations of pedal force and pedal speed to produce the same power (say, the same 300 watts) on the flat than on climbs. We would need more data to be sure but another contributing explanation to why there is a power difference between your climbing and flat land power is that perhaps you are one of those riders who can produce more power over short durations (like 5 minutes or so) at low cadence and high torque than at high cadence and low torque. This is something that cannot be answered from only two rides but it is something to keep in mind.
So, different types of races put different types of demands on the racer, and different racers can have different preferences for how to "feel most comfortable" in producing that power. If you do different types of races you may want to be sure to include training not only for your FTP or VO2Max "power" but also the pattern of pedal force and pedal speed that characterizes each type of race. As an aside, these different patterns of pedal force and pedal speed during climbing show why simply elevating the front of a bike on a trainer a few degrees does not produce the same training effect as actual climbing on a hill.