See example.
First we create calendar table - all month between min and max dates in table, or other date interval.
Then, expand status changes (status,from_date)->(status,from_date,to_date=next_date=next from_date).
Then JOIN Calendar table and expanded status changes table and get status for every Id for every possible months.

with recursive dates as(  -- calendar table
  select min(date) yy_mm_1,max(date)edt from test
  union all
  select 
--    date_add(yy_mm_1,interval 1 month) yy_mm_1  -- mySql
    dateadd(1,month,yy_mm_1) yy_mm_1            -- SQL Server
    ,edt 
  from dates
  where 
   --  date_add(yy_mm_1,interval 1 month)<=edt
     dateadd(1,month,yy_mm_1)<=edt

)
,status_changes as( -- status changes from date to next date
select * 
  ,lead(date)over(partition by id order by date) next_date
from test
)
,all_status as( -- JOIN Calendar and status_changes
select d.yy_mm_1,t1.* 
from dates d
left join status_changes t1
 on t1.date<=d.yy_mm_1 and (d.yy_mm_1<t1.next_date or t1.next_date is null)
)

-- group by year_month and status
select yy_mm_1,status,count(*) status_count
  -- ,group_concat(id order by id separator ',') ids -- MySql
  ,string_agg(id,',') within group(order by id)ids    -- SQL Server
from all_status
group by yy_mm_1,status
order by yy_mm_1

For your test data

create table test (ID int,Date date,Status varchar(10));
insert into test values
 (1,'2020-10-01','Silver')
,(2,'2021-01-01','Gold') 
,(3,'2020-11-01','Bronze') 
,(1,'2020-12-01','Bronze') 
,(1,'2021-03-01','Platinum') 
,(2,'2020-10-01','Bronze') 
,(3,'2021-04-01','Diamond') 
,(2,'2021-04-01','Bronze') 
,(1,'2020-04-01','Gold')
;

result is

For the selected report interval, you can modify the query as follows (MySql syntax)

with recursive dates as(
  select cast('2020-10-01' as date) yy_mm_1,cast('2021-02-01' as date)edt 
  union all
  select date_add(yy_mm_1,interval 1 month),edt from dates
  where date_add(yy_mm_1,interval 1 month)<=edt
)
,status_changes as(
select * 
from(
  select * 
     ,lead(date)over(partition by id order by date) next_date
  from test
  where date<=cast('2021-02-01' as date)
  )t
where next_date>=cast('2020-10-01' as date) or next_date is null
)
,all_status as(
select d.yy_mm_1,t1.* 
from dates d
left join status_changes t1
 on t1.date<=d.yy_mm_1 and (d.yy_mm_1<t1.next_date or t1.next_date is null)
)
select yy_mm_1,status,count(*) status_count
  ,group_concat(id order by id separator ',') ids
from all_status
group by yy_mm_1,status
order by yy_mm_1

There result is