Nope, it won't occur in One Shot as well.
The "fault" isn't with the camera. Sorry, but it's with the person using the camera. Please look in the manual on pages 80 and 199 to have a better understanding of the Assist Points and AF Expansion, how it works, when it doesn't, and how the invisible points are arranged.
AF Point Expansion of the type on 5DII (and 5D classic) is unique to these models and it's very important to remember that it only works in AI Servo. It does not work in One Shot. This is why you are getting better results with One Shot, shooting an angled subject in your tests.
AF Point Expansion on 5D/5DII is done with six "hidden/invisible" points clustered right around the center AF point (it's sort of as if the entire Spot Metering circle area becomes one big AF point, except that only part of it is the better "dual axis" type of point). When you are using AI Servo with Expansion on, you are risking or causing an error... see the keyboard example above. The camera will focus on whatever is closest to the camera and covered by an AF point. So one of the Expansion points below the center is focusing on a closer object, rather than the center point that's the only one active in One Shot. So, in an example like this with the subject at an angle and with Expansion turned on, the camera will focus on two different points depending upon whether it's set to AI Servo or One Shot. It will appear to "front focus" slightly in AI Servo, because the invisible point below the center one is being used. While in One Shot, only the visible center point is being used.
The only time to use Expansion Points is when shooting tough to track, moving subjects. And you will likely still have some missed focus shots, as the camera will sometimes choose the wrong point and focus on something other than what you wanted. I rarely, if ever, use Expansion on 5DII my 7Ds. I'll usually only use it when the background is a great distance from the subject or the subject is against a very plain background (such as birds in flight against the sky). By working at keeping a single point on the subject, I get better accuracy. In other words, I make the camera focus where I want it to focus, rather than leaving it up to the camera to decide.
Try turning off Expansion Points and only using the center AF point. That will give you more consistent results between the two focus modes because the camera will then function much more the same in both One Shot and AI Servo.
However, the 5DII isn't the greatest tracking moving subjects. So you should expect some misses in AI Servo, with moving subjects.
Only use the center point when shooting AI Servo, too... as it's much more sensitive than the visible peripheral points in the 5D/5DII.
Usually when I am shooting AI Servo I try to stop down a bit, to f4 or f5.6, to help improve accuracy (by increasing "forgiveness" of focus). Does it work? Well I shot nearly 2000 images last Sunday... all on AI Servo.... I've only edited about 1500 of them, but so far I've "trashed" about 15 or 20 shots for missed focus problems (I've trashed lots more for other problems). And I know a couple of those were my own fault, not the cameras'. And I know I was pushing my luck with a couple as well... shooting between obstructions or shooting really fast. (I happened to be using my 7Ds, which are much better tracking subjects... but AI Servo is AI Servo, no matter what camera we're talking about). Single point AF... center point for nearly all shots. All together, in the last six or seven weeks I've taken 17,000 shots with my 7Ds and have about 1 or 2 percent loss to missed focus. 5DII isn't quite as good, but with practice and used right can do pretty well with moving subjects (I tend to use it much more for stationary).
AI Servo looks for movement and never stops focusing... So with a static subject at times the camera will defocus and then refocus, trying to predict movement that's not happening when the subject (or photogrpaher and camera) doesn't move. So you have to expect some misses with AI Servo.... no camera is going to be perfect.
One Shot seeks focus, then when it's achieved focus, it stops and locks. It is more accurate and recommended for stationary subjects. You also get Focus Confirmation with it. One Shot cannot be used with moving subjects, unless using a pre-focus technique which is pretty limiting.
Actually, if you are looking for extremely accurate focus, switch to Live View. It's the most accurate of all... but it's quite slow.
AI Focus is pretty much worthless, sort of a snapshooter's mode. It's not really a focus mode at all. In AI Focus the camera is supposed to decide for you whether or not the subject is moving, then switch to whatever is the approptiate mode (AI Servo or One Shot). I tried working with it for a while about ten or eleven years ago (with my EOS-3s)... gave up and got far better keeper rate not using it. I haven't used it since.
So, in a nutshell... Turn off Expansion Points. It's what's causing the focus errors you are seeing. With Back Button Focusing, you should then see good results with AI Servo, but when subjects are stationary and you want to use a larger aperture, up close, with shallow depth of field where accuracy is important, I'd suggest switching to One Shot... Or Live View.