Using center point for the subject matter you describe (little action, mostly portraits) 5DII will be fine for you and feel pretty familiar. Especially if you are mostly using One Shot (I'll assume you are, since you mention focus and recompose techniques). Using a 5DII that way will be very similar.
If you were shooting sports/action, then the 60D would have an advantage, would do a better job tracking (AI Servo shooting). You'd see your "keeper rate" drop significantly, if that were the type of subject matter you were shooting. But it's not, from your description.
Before someone flames me... Yes you can use a 5DII for action shooting... You'll get some great shots with it. Just don't expect the same percentage of "keepers" if you do. With 50D or 7D I get 95% + shots acceptibly within focus. With 5DII, using the same lenses and techniques and shooting the same subjects, it drops below 50%. Now, I don't make a habit of using 5DII for sports and might get a little better with practice (some of misses are always due to the user, some to the camera... hard to really say which is which). But why force it when I have other cameras that serve the purpose better. I use the 5DII for it's strengths: gobs of fine detail, slightly greater dynamic range, lots of potential for enlargement, higher usable ISO (about a stop more than 7D/60D... two stops more than 50D), wide lenses that are truly wide, more control over depth of field and a few other reasons. I don't use it for it's weaknesses: mostly its slower AF, lower frame rate, noisier shutter (actually mirror slap), less "reach", somewhat more limited lens selection. 5DIII does address some of these very nicely... at fairly hefty big price.
Both cameras have the 9-point AF system. A key difference is that the 60D has cross type points throughout, all 9 of them, with the center one is further enhanced.
The 5DII also has 9 points in the viewfinder, but only the center one is the more sensitive cross type and roughly equal to the center point in the 60D. And even that isn't as good tracking movement,. However, it does continue to work in a bit lower light than most of the other cameras...It's slow, but can still focus about 1EV lower light than 7D, 60D, 50D, 40D. (7D is superior for moving subjects because it also has a discrete AF processor, while all the other models short of 1-Series share AF processing and image processing through a single processor). The Rebel/xxxD models that have 9 points and 30D, 20D are more similar to the AF system in the 5D/5DII. EOS-3 is radically different... it was the first iteration of the 45-point system used in 1-series cameras... very fast and configurable.
And, 5DII actually have 6 additional "hidden/assist" AF points on the 5DII, clustered close around the center point. Can't see them in the viewfinder, they are also the plain single axis type and they only work in AI Servo, but they can be enabled with a Custom Function and are intended to help shooting moving subjects. I find them to be of limited usefulness and rarely bother. I'll just work harder to keep the center point on a moving subject, if using the FF camera for moving subjects for some reason.