Will the new iPhone 7 be compatible with other wireless headphones? It was not clearly explained, whether new iPhone 7 will be only compatible with affiliated brands rather than free to connect with any wireless headset.
In a word, yes.
The iPhone 7 &amp; 7 Plus both use Bluetooth 4.2 technology.  Considering the vast majority of wireless headphones use bluetooth, this means the iPhone 7 &amp; 7 Plus will be able to pair with them just as previous iPhone models have in the past.
From Apple's iPhone 7 / 7 Plus technical specifications page, under the Cellular &amp; Wireless section:

802.11a/b/g/n/ac Wi‑Fi with MIMO
Bluetooth 4.2 wireless technology
NFC