HR measurement incorrect / to high - Fenix 5s

Sounds strange that both the watch and the HRM belt would report equally wrong increased HR. How did you measure the true HR, that you are sure the ones frome the watch and from the HRM belt are wrong?

Because it suddenly bumps up in periods with low load.

Because it suddenly bumps up in periods with low load.

That's fully normal. When you are recovering from a high anaerobic load, the body needs to eliminate the lactates quickly, hence the increased HR

I have used it for over a year, and it just started with this now around a week ago. In a 10km run, with 5:30 pace, the HR is normally and for the first kilometers beetween 140 and 150. And suddenly for the rest of the run it averages around 170 and peaks around 185 (my max) even if the load is the same, in relatively flat terrain.

Which strap are you using? I would make sure you have a fresh battery in the HR belt. I use the HRM-Run and frequently encounter this issue, especially at the start of the activity and for the first mile, my HR will appear to be in the 150-200 range, even though I'm just getting warmed up. Using electrode gel sometimes help, but in very dry (winter in my location) conditions this issue is persistent. Sometimes it helps to put in a fresh battery.  I've tested with another strap and do not get the same erroneous results. What's throwing me off is that you say it's the same with the strap or without it, assuming you mean without it, you have a HR sensor on your watch that is giving the same results. If you can, test with another belt.

and it just started with this now around a week ago

Well, if two different devices, and each with a completely different method (Optical HRM vs a HRM with eletrodes) report the same, then I'd suspect there may be a reason. If you do not trust the devices, try another one, or measure your HR manually.

Yes its the same with the hr sensor on the watch (fenix 5s). I use the hrm run 4 strap, and have used it for over a year. This problem suddenly accured. And my training condition dropped

The HRM-Run has a history of going bad within the year. I would check by replacing the battery (CR-2032). Then test by comparing to manually checking your pulse or with a different HR strap. My other strap is a Polar and is almost always accurate, though it doesn't have the running dynamics of the Garmin HRM-Run.  If still the same, contact Garmin Support. They might replace it, if still within warranty period. They did for me. However, when mine went bad, it stopped recording running dynamics, then eventually stopped connecting to the watch altogether. 

First suspected the battery so I've changed this, no difference. And as I mentioned, its the same when im not using the strap (measured with the watch)

And as I mentioned, its the same when im not using the strap (measured with the watch)

As I wote, borrow another device, or measure your HR manually, or do a test in a lab, to see what the real HR is. If it shows both your devices are wrong, have your HRM-Run replaced, as seilogramp suggests. The OHRM of the watch is not supposed to be very accurate, especially when you move, so there the wrong HR would likely not get you a new watch just for that.