You want array_contains():

where array_contains('cats'::variant, column2)

Rather than using JSON, you can use array types directly E.g.:

CREATE TABLE ts2 (obj_id INT, ts DATE, tags ARRAY, val INT);
INSERT INTO ts2 (obj_id, ts, tags, val) select 442243, '2017-01-01', ARRAY_CONSTRUCT('red', 'small', 'cheap'), 1;
INSERT INTO ts2 (obj_id, ts, tags, val) select 673124, '2017-02-01', ARRAY_CONSTRUCT('red', 'small', 'expensive'), 2;
INSERT INTO ts2 (obj_id, ts, tags, val) select 773235, '2017-01-01', ARRAY_CONSTRUCT('black', 'small', 'cheap'), 3;

The VALUES clause cannot use functions such as ARRAY_CONSTRUCT, but INSERT-SELECT will work. (You could also do this with JSON and a VARIANT type, but then you would need to tag the value with a key name, and use PARSE_JSON in the insert.)

Then to query rows that only contain exactly both tags you chose, use a query like this:

select 
  obj_id,
  tags
from ts2
where ARRAY_CONTAINS('small'::variant, tags)
  and ARRAY_CONTAINS('cheap'::variant, tags)
; 

select 
i.path, i.value::string object_keys_values
from 
cte , lateral flatten(input=>myarray) j , lateral flatten(input=>j.value) i  
where i.path <>'e';

cte2 as (select  
   object_agg(i.path, i.value::variant)  obj
from cte , lateral flatten(input=>myarray) j , lateral flatten(input=>j.value) i where i.path <>'e' 
group by  j.value)
select array_agg(obj) final_array from cte2

with cte as ( select '[email protected]'  email, 
           [     object_construct('a',1,'b','BBBB', 'c','sheep','d',1,'e','BBBB', 'f','kumera')  
               , object_construct('a',10,'b','BBBB', 'c','kiora','d',10,'e','BBBB', 'f','pavalova')
               , object_construct('a',100,'b','BBBB', 'c','gidday','d',100,'e','BBBB', 'f','kiwifruit')
           ] myarray )
 select  * from cte ;

ALTERNATIVE APPROACH -> KEEPING ORDER IN SYNC

    with cte as ( select '[email protected]'  email, 
       [     object_construct('a',1,'b','BBBB', 'c','sheep','d',1,'e','BBBB', 'f','kumera')  
           , object_construct('a',10,'b','BBBB', 'c','kiora','d',10,'e','BBBB', 'f','pavalova')
           , object_construct('a',100,'b','BBBB', 'c','gidday','d',100,'e','BBBB', 'f','kiwifruit')
       ] myarray )
       
 SELECT EMAIL, ARRAY_AGG( OBJECT_DELETE(j.value, 'e' )) from cte, 
lateral flatten(input=>myarray) j GROUP BY EMAIL;