I'm currently working as part as a 4-manperson scrum team that is responsible, to one degree or another, for everyone of our company's products. Totaling at roughly 16 products, plus a mess of semi-connected one-off's, I can tell you from experience that scrum doesn't endear itself to a multi-project environment. As stated above, it is tough to build team synergy when you're constantly working individually on different things. Furthermore, it is tough to cross-relate to the working details of your teammates' assignments, since your focus is on a completely different assignment, in a completely different project.
Moreover, 'falling-in-love' or even unassigned analysis with a particular product is nearly impossible due to the rate of assignment turnover, which can lead to code rot, among other things.
If you find yourself in a position where you can't escape multiple projects assigned to your team, I wouldn't recommend SCRUM.