Easily give role in one database access to tables in another database on the same server

Database roles are security principals that are wholly contained within their respective database and are not shared or visible to other databases. So any roles and users that are in database X have no knowledge of database Y. To accomplish your goal, you'll need to recreate the role in database Y and add all the appropriate users to this database and role. Fortunately, you can script out this process using the system views:

USE [Y];
CREATE ROLE foo;

--grant all perms on foo

--Use this generate sql to add all your users to the role
select 'CREATE USER '+quotename(u.name)+';' + char(10)+
       'EXEC sp_addrolemember ''foo'','''+u.name+''';'
from (select name,principal_id
        from x.sys.database_principals where type = 'R') r
    join x.sys.database_role_members rm 
        on (r.principal_id = rm.role_principal_id)
    join (select name,type_desc,principal_id
        from x.sys.database_principals where type != 'R') u 
        on (rm.member_principal_id = u.principal_id )
where r.name = 'foo'

You can then take this sort of script/process to a batch job to keep things synchronized, but you will need some sort of external process to keep these roles in sync.

One piece that might be confusing here is that you have logins and other server level principals that are connected to the database principals, but there is a clear distinction between these. I provide some explanation about the differences between these principals in this answer

If you work with Windows Authentication, you can add your domain users to a domain group and add this group as a login to SQL Server. Then give that login the desired permissions on the relevant databases.