I'm basing a lot of this on my own personal experience so, by all means, take it as anecdotal if you'd like.
I started exercising the week of my 34th birthday (about 8 months ago) after having smoked for 15-16 years and living a sedentary lifestyle. I discovered the heart rate zones early on, just like you, and I got my heart rate monitor (FitBit Charge HR) as well.
Some people will say that if you want an accurate heart rate, VO2 measurement, etc., you'll need to go see a medical professional, have a bunch of gear hooked up, and so on and so forth. They're right.
That being said, I have measured my FitBit heart rate against probably 20-30 different types of cardio machines (treadmills, bikes, stair stepper, etc...) from different manufacturers, and my heart rate on the Fitbit almost always matches the heart rate on the machine within 1-2 beats.
Considering that the FitBit uses a different methodology than ordinary cardio equipment, I feel like it's a decent enough approximation. Might I see a difference between my FitBit and a professional medical device? Sure. Is it going to be significant enough to where one day you find out you've never even hit zone 5? I highly doubt it.
Furthermore, the FitBit uses my heart rate to determine total caloric burn. Again... not a medical device, but considering that I've lost 35 pounds by studiously trying to keep to a -500 calorie daily deficit, and been successful with it, my guess is the FitBit is accurate enough to be useful.
So, that being said, the question is whether or not exercising at that heart rate is dangerous. Obviously, each individual is different, so it's not a one-size-fits-all answer. That being said, I imagine if you told your doctor you were doing this, they'd probably jump for joy. The first few weeks/months of exercise for me, as soon as I took a step, I was virtually in zone 5. Now... I have to push it very hard to get there. So, I somewhat suspect that the reason you're always in this zone is because you're once again challenging your body.
So, my guess is that, over time, and especially if you don't vary your routine, your heart and the rest of your cardiovascular system will learn to very quickly adapt to this exercise routine. As a result, you'll probably struggle to get into that range as time goes on.
For normal people, I haven't read that this is overly problematic. That being said, if you have an underlying or undiagnosed condition, I suppose it could be dangerous. One thing I noticed early on was that if I overtrained, the next day I might have a very mild heart flutter. It may have only happened for 2-3 seconds out of the entire day. It didn't happen all the time, and it was usually when I was at rest, but I knew that it was time for me to take an extra day off, and then I'd be fine.
Obviously, you know how you feel more than any of us do. I'd say that if you're feeling great, then you're doing fine. The more you do it, the better you'll feel. I'm at a point now where missing my gym days puts me in a bad mood. I look forward to it. I love it.
I like the heart rate monitor. I have learned its intricacies, it allows me to gauge my current performance against past performance, and although I can say that I am responsible for my weight loss and lifestyle change, the FitBit was an excellent enabler.