As an owner of Disney Infinity on three major console platforms, the answer is no, because of Xbox 360.
The Xbox 360 Infinity Base has to work with Microsofts USB Device security chip Infineon, Xbox 360 negotiates with USB devices to prove they are authorised peripherals (by Microsoft for use on Xbox 360), so the PS3 and Wii U (and Wii) Infinity Bases do not work on Xbox 360 as they do not include this extra communication and most likely additional electronics inside, only the Xbox 360 one does, furthermore the extra negotiation in the 360 base causes it to come up as an "Unrecognised USB Device" (on PS3), when attached to a different console than an Xbox 360, and does not work on anything but an Xbox 360.
It actually says on the underside "for Xbox 360" on the one in the Xbox 360 Starter Pack, which helps to not confuse them, whereas the PS3 and Wii U appear mostly identical.
I have been able to use the Wii U and PS3 Infinity bases interchangeably on each platform/format.
I do not have the Wii version, however I would imagine it is the same as the PS3 and Wii U.
As long as Xbox 360 is not one of your platforms/formats then you'll be able to share/reuse the base.
This presents an interesting potential problem, if Microsoft decide to prevent Xbox 360 accessories from working on Xbox One or do something different to Infineon on Xbox One, then a new base will be required for an Xbox One Disney Infinity release.
See more insight on Multi-Format for Disney Infinity here:
Multi-format Disney Infinity Perspectiveviewtopic.php?t=3210#p33613