Well this was a welcome comment @foxfireorchard! Thank you for taking a genuine interest. And for giving me such a huge compliment!
In answer: yes the process is exactly the same. The difference will be in the conversion rate. So while my 10 litres of goats milk makes a 700g Cotija your 10 litres of cow milk should covert to about 1.2kg cheese - depending on varying factors. Obviously the taste is different. The stage where you are stirring the curds may take longer. I prefer vegetable rennet. Many reasons. The difference is not in microbial versus animal but more the manufacturers. I pay more but get from a reputable importer who gets certified non GMO cultures and rennets from Italy. Same applies to the culture. You get mesophilic and thermophilic. Which is important if you are a master cheesemaker! For the rest get a mix or rather stick with mesophilic.
Have you ever made cheese before? I started about a decade ago making peasant cheese (my word). It is really easy and doesn't follow the strictest cheese making rules - but it works. Most of the time. If you get the milk still warm that is best. If not warm it slightly. Then either use lemon juice or apple cider vinegar. While stirring the milk slowly drizzle in your acidity of choice. We have a lemon tree and I like the citrus taste in cheese so that is my choice. It roughly works out (with goats milk) to one lemon per litre. How you will know if and when the job is done; the milk will suddenly curdle. STOP pouring ACV or lemon when that happens. Scoop out the curds and use immediately for a really soft peasant cheese. Or hang up in muslin cloth. I used to often toss in black pepper and sundried tomato or olives. Also fresh herbs. YUMMY!
Please let me know when you make the cheese.