Not sure what military installations UHBlackhawk has been stationed at, but I saw quite a few bumper stickers, mostly conservative, at my last base and even more at my current station. Also, as an AD officer I have never heard of any "tradition" of not registering with political parties. I know many officers, both AD and retired, who have done so.
Again, I've heard many a servicemember talk politics, both behind closed doors and in the office. Granted, it was all done respectfully for the most part, but it still occurred. I will agree with UHBlackhawk that is not common to see servicemembers who are far apart in rank or officers/EM's to discuss politics together, but it really isn't that uncommon for young officers and young EM's to discuss politics among their peers.
I think the recent election season saw a few changes, whether they be temporary or permanent, regarding attitudes toward politics in the military. Was it because we had a candidate who was caught keeping classified info on an unclassified medium (something servicemembers would be court-martialed for)? Maybe. Was it because some thought that President Trump was unfit to lead the nation's military? Maybe. But I think you will find that some political issues can't just be "not talked about" in the service, especially those that deal with readiness issues such as transgender service, or deployment issues such as how many troops politicians may or may not send overseas, etc.
To actually answer the OP's question, you will find that the military does typically lean to the right (past surveys of military members usually corroborate this). It's just the way it is, but even if you are a flaming liberal (and I'm not saying you are) I'm sure you'll be fine. I wouldn't exactly advertise this fact among my unit if I was you, but then again I wouldn't advise anyone in the service no matter what their party was to go around telling everyone what edge of the spectrum they fall into. And if it makes you feel any better, while all the services tend to lean right to a certain degree, the Navy and the Air Force usually have held the least amount of conservative tendencies among their personnel, while the Army and Marine Corps are known to be significantly more conservative.