You most definitely can study music production as a degree. I completed a BA Honours 'Creative Music Production' degree accredited to Surrey University around 2012.
If you are in the UK there are many music production degrees available. ACM, BIMM and SAE are all special institutes where you can study a music production degree. There are also some universities that offer this, though be wary that some may not have amazing studio facilities.
You can also do online degrees from places like PointBlank. Or you can also study at a lower level and do an HNC or HND. There are also variations such as audio engineering. Though be careful of any called 'Sonic Arts' as these focus on very abstract music.
If you are in the US there are many institutions offering music production majors, such as the world famous Berkley, offering a course in music production and engineering. I'm sure there are many others that a google search will reveal.
On a personal note - studying music production at ACM (Academy of Contemporary Music) was the first time I'd really enjoyed studying anything. I'd say it is an extremely interesting and a practical / hands on degree. Learning to record bands and work in a studio environment was a fantastic experience that I wouldn't change.
However, be aware that the music production degree won't open too many doors for you once you leave university. It helped me get my first jobs in the music industry, but outside of this industry it won't carry too much weight.
Another factor to consider is there are not too many large recording studios out there so you will be competing with a lot of people for very few opportunities in the field.
The final thing to consider is whether you can afford to pay back a large student debt - I still am paying off mine 10+ years later. This can be stifling. With other modern education -like online courses and online tuition there are certainly cheaper, more personalised alternatives out there.