Site similar to fccid.io for Europe
There is not a similar site for Europe, because the US and EC does not handle EMC the same way. You can read more about the difference in The accepted answer to this question.
TL;DR, there is no central register for Electromagnetic compatible products/manufacturers.

But, you state that you use the database as a resource to figure out what frequencies are used by what systems - then there is a register for that, of course. There are organizations that govern the use of frequency bands. Then it will be CEPT (The European Conference of Postal and Telecommunications) you want.
Specifically, The European table of frequency allocations and applicatons in the frequency range 8.3 kHz to 3000 GHz (ECA TABLE)

Bear in mind that all bands are not necessarily harmonized in all countries. To be able to know what frequency bands are being used for what systems in what country, you will have to refer to the local Postal and Telecommunication agency in each separate country.