I was able to solve the problem by adding two additional URLs to Kee.
That is, my KeePass entry’s URL is set to https://banking.ing-diba.de, so there is 3 URLs in total now.
In addition to the above URL I added the following ones for each KeePass entry (I have several accounts) on the Kee-tab/URL-tab:
https://access.ing.de
https://banking.ing.de
In all tests I’ve made so far Kee always presented the matching KeePass entries.