Introduction
Nike
Adidas

Adidas promotes its 'own' shirt numbers, as far as I know, since 2000. Unlike Nike, Adidas doesn't introduce new numbers every two year. Untill now, May 2005, she sticks to the same typeface: Serpentine Medium Oblique. This is a commercial available typeface, and as far as I can judge the only thing that has been altered in the design is that the numbers are electronically condensed and stroke widths are slightly adjusted. From a standpoint of branding and protection against plagiarism this is a doubtful choice. In fact everyone can buy this typeface, condense it and use it to print names and numbers on a shirt without violating copyrights.
Another difference with the Nike approach is that Adidas seems to be less compulsory towards teams playing in their kit. As can be read in the teams' section, Ajax and Real Madrid —both wearing Adidas shirts— have their own specific shirt numbers.

Figures from Serpentine Medium Oblique, squeezed to 60 percent.