But I could literally not find a single 5GHz amplifier. Why is that? Am I just too stupid to find them or is it simply not possible to amplify 5Ghz in the same fashion for some reason? If so, why?
On the very page you linked to, when I scrolled way down to the "More Products" section of the page, I found this item which is similar to the product you linked. Granted, it is in a lot of 10, but I am sure a bit more searching on the site will produce single units.
But to answer more of the heart of your question, any signal can be amplified no matter the frequency. By stating that, I don't intend to simplify this issue as it is much more complex.
Namely, 802.11 is two way communication. Boosting/amplifying the signal is generally done on the transmit side to increase the transmit power. However they are often designed not to amplify the the received power. Why? Because they would have to try to filter out the non-802.11 signals before amplifying the signal or they would simply amplify the noise as well. Since signal quality is what you are really after, boosting the signal and the noise an equal amount doesn't improve the signal quality at all.
Filtering out non-802.11 signals and other sources of interference is more complicated. So generally when you are using a signal amplifier, you are expected to use one on both sides. When you have two devices transmitting at significantly different power levels, you can experience a number of issues due to much lower data rates in one direction or complete lack of signal in one direction.
While it has been mentioned, I must also point out that most areas of the world have some sort of organization/government body that oversees the use of the RF spectrum in their jurisdiction. Make sure you adhere to local laws as you may have legal/financial repercussions if you do not.
Additionally, make sure whatever you are doing is "friendly" to any neighbors as well. If you affect others and they discover that it was your fault for impacting something they expect to have available to them, you never know how they will react. It's overly extreme, but people shoot other people for cutting them off on the expressway, and many would consider breaking their Internet access a worse offense.
Going beyond those issues, 802.11 design today is generally no longer as much about "coverage" as it is about "capacity." Any sort of amplifier/extender/repeater gains an increase in coverage at the potential loss of capacity. The trend is lower power to produce more smaller "cells" to allow for more client devices and more bandwidth per device. Just because you can make a cell bigger, doesn't mean that you should. The demand for such devices is decreasing, so expect them to become rarer over time.