UPDATE:
I did some changes, because I’d forgot to involve „Public Folders“, and „Distribution Lists“. This time I use not only two, but three separated mail databases, bud grouping two of them together and one is completely separated:
New-AddressList -Name "All Rooms (DOM1+DOM2)" -RecipientFilter {(Alias -ne $null) -and ((Database -eq "CN=DOM1,CN=Databases,CN=Exchange Administrative Group (FYDIBOHF23SPDLT),CN=Administrative Groups,CN=First Organization,CN=Microsoft Exchange,CN=Services,CN=Configuration,DC=example,DC=com") -or (Database -eq "CN=DOM2,CN=Databases,CN=Exchange Administrative Group (FYDIBOHF23SPDLT),CN=Administrative Groups,CN=First Organization,CN=Microsoft Exchange,CN=Services,CN=Configuration,DC=example,DC=com")) -and (((RecipientDisplayType -eq 'ConferenceRoomMailbox') -or (RecipientDisplayType -eq 'SyncedConferenceRoomMailbox')))}
New-AddressList -Name "All Rooms (DOM3)" -RecipientFilter {(Alias -ne $null) -and (Database -eq "CN=DOM3,CN=Databases,CN=Exchange Administrative Group (FYDIBOHF23SPDLT),CN=Administrative Groups,CN=First Organization,CN=Microsoft Exchange,CN=Services,CN=Configuration,DC=example,DC=com") -and (((RecipientDisplayType -eq 'ConferenceRoomMailbox') -or (RecipientDisplayType -eq 'SyncedConferenceRoomMailbox')))}
New-AddressList -Name "All Users (DOM1+DOM2)" -RecipientFilter {(Alias -ne $null) -and ((Database -eq "CN=DOM1,CN=Databases,CN=Exchange Administrative Group (FYDIBOHF23SPDLT),CN=Administrative Groups,CN=First Organization,CN=Microsoft Exchange,CN=Services,CN=Configuration,DC=example,DC=com") -or (Database -eq "CN=DOM2,CN=Databases,CN=Exchange Administrative Group (FYDIBOHF23SPDLT),CN=Administrative Groups,CN=First Organization,CN=Microsoft Exchange,CN=Services,CN=Configuration,DC=example,DC=com")) -and (((((((ObjectCategory -like 'person') -and (ObjectClass -eq 'user') -and (-not(Database -ne $null)) -and (-not(ServerLegacyDN -ne $null)))) -or (((ObjectCategory -like 'person') -and (ObjectClass -eq 'user') -and (((Database -ne $null) -or (ServerLegacyDN -ne $null))))))) -and (-not(RecipientTypeDetailsValue -eq 'GroupMailbox'))))}
New-AddressList -Name "All Users (DOM3)" -RecipientFilter {(Alias -ne $null) -and (Database -eq "CN=DOM3,CN=Databases,CN=Exchange Administrative Group (FYDIBOHF23SPDLT),CN=Administrative Groups,CN=First Organization,CN=Microsoft Exchange,CN=Services,CN=Configuration,DC=example,DC=com") -and (((((((ObjectCategory -like 'person') -and (ObjectClass -eq 'user') -and (-not(Database -ne $null)) -and (-not(ServerLegacyDN -ne $null)))) -or (((ObjectCategory -like 'person') -and (ObjectClass -eq 'user') -and (((Database -ne $null) -or (ServerLegacyDN -ne $null))))))) -and (-not(RecipientTypeDetailsValue -eq 'GroupMailbox'))))}
New-AddressList -Name "Public Folders (DOM1+DOM2)" -RecipientFilter {(Alias -ne $null) -and ((Database -eq "CN=DOM1,CN=Databases,CN=Exchange Administrative Group (FYDIBOHF23SPDLT),CN=Administrative Groups,CN=First Organization,CN=Microsoft Exchange,CN=Services,CN=Configuration,DC=example,DC=com") -or (Database -eq "CN=DOM2,CN=Databases,CN=Exchange Administrative Group (FYDIBOHF23SPDLT),CN=Administrative Groups,CN=First Organization,CN=Microsoft Exchange,CN=Services,CN=Configuration,DC=example,DC=com")) -and (ObjectCategory -like 'publicFolder')}
New-AddressList -Name "Public Folders (DOM3)" -RecipientFilter {(Alias -ne $null) -and (Database -eq "CN=DOM3,CN=Databases,CN=Exchange Administrative Group (FYDIBOHF23SPDLT),CN=Administrative Groups,CN=First Organization,CN=Microsoft Exchange,CN=Services,CN=Configuration,DC=example,DC=com") -and (ObjectCategory -like 'publicFolder')}
New-AddressList -Name "All Contacts (DOM1+DOM2)" -RecipientFilter {(Alias -ne $null) -and (RecipientContainer -eq "OU=DOM1\+DOM2,OU=Contacts,OU=COMPANY,DC=example,DC=com") -and (((ObjectCategory -like 'person') -and (ObjectClass -eq 'contact')))}
New-AddressList -Name "All Contacts (DOM3)" -RecipientFilter {(Alias -ne $null) -and (RecipientContainer -eq "OU=DOM3,OU=Contacts,OU=COMPANY,DC=example,DC=com") -and (((ObjectCategory -like 'person') -and (ObjectClass -eq 'contact')))}
# tried this for distribution lists, but to solve it as quick as possible I decided to create it in EMC; this doesn't work, but it can be useful for notion:
#New-AddressList -Name "All Distribution Lists (DOM1)" -RecipientFilter {(Alias -ne $null) -and (ObjectCategory -like 'group') -and (RecipientContainer -eq "OU=DOM1,OU=Groups,OU=COMPANY,DC=example,DC=com")}
#New-AddressList -Name "All Distribution Lists (DOM1)" -Container "example.com/COMPANY/Groups/DOM1" -RecipientFilter {(Alias -ne $null) -and (ObjectCategory -like 'group')}
New-GlobalAddressList -Name "Global Address List (DOM1+DOM2)" -RecipientFilter {(Alias -ne $null) -and ((Database -eq "CN=DOM1,CN=Databases,CN=Exchange Administrative Group (FYDIBOHF23SPDLT),CN=Administrative Groups,CN=First Organization,CN=Microsoft Exchange,CN=Services,CN=Configuration,DC=example,DC=com") -or (Database -eq "CN=DOM2,CN=Databases,CN=Exchange Administrative Group (FYDIBOHF23SPDLT),CN=Administrative Groups,CN=First Organization,CN=Microsoft Exchange,CN=Services,CN=Configuration,DC=example,DC=com")) -and (((ObjectClass -eq 'user') -or (ObjectClass -eq 'contact') -or (ObjectClass -eq 'msExchSystemMailbox') -or (ObjectClass -eq 'msExchDynamicDistributionList') -or (ObjectClass -eq 'group') -or (ObjectClass -eq 'publicFolder')))}
New-GlobalAddressList -Name "Global Address List (DOM3)" -RecipientFilter {(Alias -ne $null) -and (Database -eq "CN=DOM3,CN=Databases,CN=Exchange Administrative Group (FYDIBOHF23SPDLT),CN=Administrative Groups,CN=First Organization,CN=Microsoft Exchange,CN=Services,CN=Configuration,DC=example,DC=com") -and (((ObjectClass -eq 'user') -or (ObjectClass -eq 'contact') -or (ObjectClass -eq 'msExchSystemMailbox') -or (ObjectClass -eq 'msExchDynamicDistributionList') -or (ObjectClass -eq 'group') -or (ObjectClass -eq 'publicFolder')))}
New-OfflineAddressBook -Name "Offline Address Book (DOM1+DOM2)" -AddressLists "Global Address List (DOM1+DOM2)"
New-OfflineAddressBook -Name "Offline Address Book (DOM3)" -AddressLists "Global Address List (DOM3)"
New-AddressBookPolicy -Name "AddressBook Policy (DOM1+DOM2)" -AddressLists "\All Users (DOM1+DOM2)","\All Contacts (DOM1+DOM2)","\All Distribution Lists (DOM1+DOM2)","\Public Folders (DOM1+DOM2)" -OfflineAddressBook "\Offline Address Book (DOM1+DOM2)" -GlobalAddressList "\Global Address List (DOM1+DOM2)" -RoomList "\All Rooms (DOM1+DOM2)"
New-AddressBookPolicy -Name "AddressBook Policy (DOM3)" -AddressLists "\All Users (DOM3)","\All Contacts (DOM3)","\All Distribution Lists (DOM3)","\Public Folders (DOM3)" -OfflineAddressBook "\Offline Address Book (DOM3)" -GlobalAddressList "\Global Address List (DOM3)" -RoomList "\All Rooms (DOM3)"
Get-Mailbox -resultsize unlimited | where {$_.Database -eq "DOM1"} | Set-Mailbox -AddressBookPolicy "AddressBook Policy (DOM1+DOM2)"
Get-Mailbox -resultsize unlimited | where {$_.Database -eq "DOM2"} | Set-Mailbox -AddressBookPolicy "AddressBook Policy (DOM1+DOM2)"
Get-Mailbox -resultsize unlimited | where {$_.Database -eq "DOM3"} | Set-Mailbox -AddressBookPolicy "AddressBook Policy (DOM3)"