XboxOne really supports 5Ghz?

This Support page about wireless network lists solutions and this information that may assist you.

Step 4: Some third-party headsets can cause wireless interference

If your router uses the 2.4GHz or 5GHz band, your wireless headset may be causing wireless interference. Typically, this occurs if you use a wireless headset with a base station (transmitter).

To continue using your wireless headset, try moving the base station (transmitter) out of the line of sight between your Xbox One and your router. Visualise a straight invisible line between your console and the router and then move the headset as far away
from your Xbox as possible and try not to overlap the line of sight between the base station and where you will be positioned while wearing the headset. Then, test your connection again. You can repeat this process until you find a good position for your headset’s
base station.

Note If your issue returns after you plug the headset back in, you may not be able to use it with your Xbox One and connect wirelessly.

Here is a chart for the minimum speeds needed for gaming and streaming movies. (website link at bottom)

Think of it this way on your computer if you are going to speedtest.net it looks for the closest server to test off of normally not to far away.

If you are doing the speedtest through the consoles network settings they are more than likely pinging the servers in Seattle WA (MS HQ) so depending on how far away you are from Seattle you are more than likely going to see a slower connection speed than
you would going on your computer that is looking for servers in your local area.

But like what was previously pointed out your speeds are well above the minimum to be able to play online according to the chart below.

Link for minimum speeds

[table][tr][td] [/td][td]Online gaming[/td][td]SD video streaming[/td][td]HD video streaming[/td][/tr][tr][td]Download speed[/td][td]3 MB/s or 3,000 KB/s [/td][td]1 MB/s or 1,000 KB/s [/td][td]3.5 MB/s or 3,500 KB/s [/td][/tr][tr][td]Upload speed[/td][td]0.5 MB/s or 500 KB/s [/td][td]N/A [/td][td]N/A [/td][/tr][tr][td]Ping[/td][td]Less than 150 milliseconds [/td][td]Less than 150 milliseconds [/td][td]Less than 150 milliseconds [/td][/tr][/table]

You have to remember when you speedtest from your xbox you are speed testing to the xbox servers. Speeds vary with speed tests depending on servers that you ping. Distance and quality of the servers can make a big difference in what a speedtest will show.
Not sure what you are playing to make you think your connection isn't fast enough. You have more than enough speed. Heck you could stream an HD movie from another device and still be playing online with your xbox and you would still have plenty of bandwidth
left.

The question is I connect to the computer and my xbox, they are in the same place next to the modem and there is a lot of difference makes me think that it does not support it.

Computer: ping in 5ms Downlaod: 150mps & upload 15mbps

XBOX: ping at 16ms Down: 80mpbs and Up: 13mbps

It is too slow to play. How can I improve the speed?

Is it a FritzBox 7490 with firmware 6.51?

If you console and PC are "next to the modem" why you aren't using a couple of ethernet cables?

Nothing is better than ethernet connection.

80mps is not to slow to play...