Yes. The Eve app and Eve accessories use Apple's HomeKit technology which requires iCloud. An iCloud account is necessary for all HomeKit apps and devices from other vendors.
There is only one way, which is absolutely NOT supported by Eve, to use individual Eve accessories with one single, local iOS device and Bluetooth, without iCloud. To do this, you must sign off from iCloud in Settings.
If your iOS device has any kind of iCloud service enabled (such as family sharing, syncing of calendars, contacts, reminders, iCloud Drive) set up on your iOS device, you must enable iCloud and iCloud Keychain for HomeKit to work properly.
Once you have any service that is part of iCloud (such as family sharing, syncing of calendars, contacts, reminders, iCloud Drive) set up on your iOS device, iOS assumes that you might want to use your data, including HomeKit data, on another iOS device as well.
In order to securely access any HomeKit accessory from any vendor, the iOS device needs a secure access token that gets negotiated as a result of entering the HomeKit setup code when pairing, during initial device setup.
The Keychain and the iCloud Keychain represent system wide storage for private data like passwords, credit card data, access tokens, and more. The iCloud Keychain also a means to securely share those HomeKit access tokens across your iOS devices. This security is all built into iOS, with no user or developer configurable options.
If you are concerned about entrusting your home configuration to iCloud, Apple has documented the complete iOS security architecture in a guide which is available here that includes information about iCloud security.