Intel I7 fit I5 motherboard

Solution
All that matters is if the socket is the same as the processor have the right BIOS version.
The exception is the new Intel 8xxx processors need a z370 board.
Mainstream boards(socket 115x) support celerons, pentiums, i3s, i5s and i7s(sockets and chipsets do have to be compatible).

With enthusiast level boards socket 1366, 2011, 2011-3 use i7s, Socket 2066 uses i7s, the new i9s, and 1 model of i5(the i5-7640X). Sounds to me like you have an enthusiast board. Some of these also support certain Xeons cpus.
 


Indeed, it might confuse.
So, the golden rule would be:
Z170 boards work with 6th gen CPUs and 7th gen CPUs after a BIOS update
Z270 boards work with both 6th gen and 7th gen Intel CPus, no matter the BIOS
Z370 boards work with 8th gen CPUs only.