Is it possible to PIVOT on a LIKE statement

SUM(CASE

For a limited number of Names you can use a SUM(CASE solution in this way:

SELECT 
    'Database status' as [DB Status],
    SUM(CASE WHEN Name LIKE 'Test%' THEN 1 ELSE 0 END) As Test,
    SUM(CASE WHEN Name LIKE 'Prod%' THEN 1 ELSE 0 END) AS Prod,
    SUM(CASE WHEN Name = 'Migrated' THEN 1 ELSE 0 END) AS Migrated,
    SUM(CASE WHEN Name = 'Offline' THEN 1 ELSE 0 END) AS Offline,
    SUM(CASE WHEN Name = 'Reserved' THEN 1 ELSE 0 END) AS Reserved
FROM 
    [Status];

PIVOT

If there is an extensive list of Names but only few of them must be rewritten you can maintain the PIVOT solution:

SELECT 'Database Status' AS [DB Status],
[Test], [Prod], [Migrated], [Offline], [Reserved]
FROM
(
    SELECT 
        ID, 
        CASE
            WHEN Name LIKE 'Test%' THEN 'Test'
            WHEN Name LIKE 'Prod%' THEN 'Prod'
            ELSE Name
        END AS Name
    FROM 
        [Status]
) AS Source
PIVOT
(
    COUNT(ID) FOR Name IN ([Test], [Prod], [Migrated], [Offline], [Reserved])
) AS PivotTable;

db<>fiddle here

DYNAMIC QUERY

If you feel a bit lazy and don't want to write all column names, you can use a dynamic query:

DECLARE @cols nvarchar(max);

SET @cols = STUFF((SELECT DISTINCT ',' + QUOTENAME(CASE WHEN Name LIKE 'Test%' THEN 'Test'
                                                    WHEN Name LIKE 'Prod%' THEN 'Prod'
                                                    ELSE Name END)
                   FROM [Status]
                   FOR XML PATH(''), TYPE).value('.', 'NVARCHAR(MAX)'), 1, 1, '');

DECLARE @cmd nvarchar(max);

SET @cmd = 
'SELECT ''Database Status'' AS [DB Status],' + @cols + ' FROM
    (SELECT 
        ID, 
        CASE
            WHEN Name LIKE ''Test%'' THEN ''Test''
            WHEN Name LIKE ''Prod%'' THEN ''Prod''
            ELSE Name
        END AS Name
    FROM 
        [Status]
) AS Source
PIVOT
(
    COUNT(ID) FOR Name IN (' + @cols + ')
) PVT'

EXEC(@cmd);

db<>fiddle here

I think it's important to strictly separate the two tasks you're trying to perform in one step here.

1. Classification
2. Transformation

For classifying the data, my instinct here is to recommend a lookup table to rigorously map records to a parent class. e.g.

CREATE TABLE StatusType (
  ID     INT         IDENTITY PRIMARY KEY,
  [Name] VARCHAR(10) NOT NULL UNIQUE
);
GO
ALTER TABLE [Status] 
  ADD StatusTypeID INT NOT NULL 
    DEFAULT 1
    FOREIGN KEY REFERENCES StatusType (ID) ;

...where the seed record in StatusType (ID=1 for the Status.StatusTypeID default) is a placeholder record named "Unknown" or similar.

When the lookup data is seeded and base records are updated with the correct keys, you can pivot to your heart's content.

select 'Database Status' AS [DB Status],
    [Test], [Prod], [Migrated], [Offline], [Reserved]
from (
    select s.ID,
           st.Name as StatusTypeName
    from status s
    join statusType st on st.ID = s.StatusTypeID
) as Source
pivot (
    count(ID) for StatusTypeName in ([Test],[Prod],[Migrated],[Offline],[Reserved],[Unknown])
) as pvt;

Full dbfiddle