Please don´t comit the sin of confusing Spanish Chorizo with Mexican Chorizo (or any other Latinamerican chorizo (Argentinean, Chilean, Guatemalan, Honduran, etc.) Besides the fact that there are more than a dozen varieties of Spanish Chorizo (fresh, cured, smoked, semi-dry, dry-cured, plus regional, ethnic, seasonal, and other varieties). A true Spanish Chorizo aficionado should know the difference between Butifarra Blanca, Soria, Cantimpalo, Herradura, Chistorra, Butifarra Anisada, Morcia, Cervelat, Sobresata, Vela, etc. (Yes I know some of these are semi-dry, some dry, etc.)
Yes, Mexican oregano is different than Greek oregano. There are also a couple varieties of Mexican chorizo. Some hot, some just spicy, fresh, cured, smoked, etc. That "Mexican" chorizo recipe Chuckwagon posted looks more like Spanish due to the large proportion of paprika. Mexican varieties have a redish color from Anatto (locally called Achiote) or chili peppers.
Centralamerican chorizos tend to be fresh and centered around vinegar, cumin and pepper.
David