That's not necessarily true, I know that you can learn how to use a katana without any other other MA training, there's swordsmanship classes, quick draw, as well as just plain old kendo and I'm not sure that the former two teach any or many open hand techniques and kendo certainly wouldn't. Even still, learning how to use your opponent's size and weight against them tends to be relegated to specific arts, not all teach that and fewer still teach ground combat. Most Asian martial arts focus on fighting on standing up and, generally speaking, are at a sever disadvantage on the ground against a judoka or someone trained in jiu jitsu. Trust me when I tell you that's a whole different game when you're on the ground wrestling with an opponent; I've been taking taekwondo for going on 3 years now and I've done a little jiu jitsu during that time and it's pretty tough and even tougher when you're going against someone bigger and stronger and knows what they're doing.
The other thing to consider is that most martial arts are not really for self defense and have evolved into more of a sport, this is especially true for any form of competitive martial art. You can be a high level black belt in a martial art but that doesn't necessarily mean that you're a trained killer. There's certain things that you do in competition and practice to keep from hurting your opponents but might run contradictory to what you'd do in a real no holds barred fight. This doesn't mean that you wouldn't be effective, just that it doesn't automatically make you an unarmed killing machine. Now something like krav maga or Marine Corps MCMAPS are something differently entirely and are not designed as sport and would be much more effective on the street or in a knock down, drag out fight with a mad Governor.
This is not intended to brag or inflate my own ego, accepting that this is mainly a prop forum and we are discussing a fictional scenerio, but a quick CV seems to be warranted.
Here's a quick rundown on my realworld expertise, 10+ yrs Army & USAF (special training in CBRNE, HAZMAT, HAZWOPR, Incident Command, EOD, beddown & SERE), 15+ yrs western martial arts specifically melee weapons from 1066-1605AD, 20+ yrs asian martial arts (TKD, Judo, Aikido, Karate, JKD & kickboxing) with 3 blackbelts and instructor certifications, 18+ yrs various asian weapons (including the katana & wakizashi).
With that in mind and sincerely not trying to instigate anything, there's not much in your previous statement I can agree with.
I've never heard of "quickdraw" so I can't comment, but I believe I said classical (or maybe traditional) swordmanship ... If not then I intended to but failed to proof read. So any modern interpretations of traditional sword training programs, IMO, don't do the student service by negating h2h skills. Kendo no more teaches how to use a real sword than Mario kart teaches how to legally drive. Kendo use shinais/bokkens and a very formalized "fight" much like modern fencing. Both would be better than nothing in TWD, but poor substitutes for real training.
The Gov (as far as we know) has no formal or otherwise training. He's just a average middle class American that can be charming. Michonne is portrayed as a trained & proficient swordsman and it can be easily inferred that she has had other combat training from her actions so far. They predominantly fought standing up (advantage Michonne). It was only after he shoved her head into the aquarium that it went to the ground, and even then it only lasted another few seconds before she shoved the glass in his eye. The several minutes prior to the aquarium she fought very poorly. I was disappointed because that's the only thing I expected her character to do well.