I would like to make 'Mexican Vanilla' extract. I have used it for years and now would like to make it.
You can find many instructions on the web for how to make your own vanilla extract, such as this one from Beanilla. As Derobert mentions, it is essentially just soaking the beans ins spirits long enough for the flavors to infuse. The beans you buy will determine the flavor profile. To make Mexican vanilla extract, you would buy pods imported from Mexico.
Regarding commercial products:
If you are speaking of pure, genuine vanilla extract of the type legal for import in the US, then Mexican vanilla extract is the same as any other.
Such vanilla is subject, in the US at least, to the FDA standard for identity for vanilla extract, and so cannot vary from this model. It is made, usually, by soaking vanilla beans in an alcohol/water solution, although there are alternate methods permitted. Several minor ingredients such as sugars or glycerine are permitted, but if used, must specified on the label.
On the other hand, there are products available in Mexico that are not true vanilla. These are made from extracts of the tonka plant, and contain a substance, coumarin, that is not permitted in foods in the US, as it is a blood thinner and can cause bleeding. The FDA says:
Be wary about buying products labeled "Vainilla" or "Extracto de Vainilla" in Mexico and other Latin American countries. Look for "vanilla bean" in the ingredient list on the label. If it has "tonka bean" or if there is no ingredient list or a vague one, do not purchase this product.