Database Cache Memory in Performance Monitor drops down significantly after DBCC CheckDB
We have been monitoring some SQLServer: Memory Manager's metrics, and
  noticed that after DBCC CheckDB job, metric
  
  Database Cache Memory (KB) drops down significantly. If to be exact,
  it dropped from 140 GB cached DB memory to 60 GB


This is correct, you can clearly see this behaviour when this example DBCC CHECKDB command completes at 21h45






Why

This behaviour is due to the database snapshot created by the DBCC command being dropped, removing all of its objects in memory.

You can replicate the behaviour by creating a snapshot of a database, loading some data in memory, and then dropping that snapshot

  CREATE DATABASE MY_DATABASE
     GO
     USE MY_DATABASE 
     GO
     CREATE TABLE dbo.bla(id int identity(1,1) PRIMARY KEY NOT NULL,
                          val int,
                          val2 char(100));



    INSERT INTO dbo.bla(val,val2)
    SELECT ROW_NUMBER() OVER (ORDER BY (SELECT NULL)),'bla'
    FROM master..spt_values spt
    CROSS APPLY master..spt_values spt2;
    GO

    CREATE DATABASE MY_DATABASE_SNAPSHOT
     ON  
     (  
     NAME ='MY_DATABASE',  
     FILENAME ='D:\DATA\MY_DATABASE.ss'  
     ) 
     AS SNAPSHOT OF MY_DATABASE;

     GO


    USE MY_DATABASE_SNAPSHOT
    GO
    SELECT * FROM dbo.bla;

     SELECT
      COUNT(file_id) * 8/1024.0 AS BufferSizeInMB
    FROM sys.dm_os_buffer_descriptors;


BufferSize before dropping the snapshot 

BufferSizeInMB
1061.70312  --before


Dropping the snapshot

USE master
GO
DROP DATABASE MY_DATABASE_SNAPSHOT ; 


BufferSize after dropping the snapshot 

BufferSizeInMB
824.179687 --after





  Second question is why "buffer cache hit ratio" did not change after
  DBCC CheckDB completes ?


This depends on how fast the data is loaded back in your buffer cache. 

If your buffer pool fills up over a longer time, it should amount to this ratio staying higher on average.

This corresponds with this part of your question:


  ... It (Buffer pool datasize) dropped from 140 GB cached DB memory to
  60 GB. and after that, slowly ramp up again during the week...