You can install the official Microsoft extension Database Administration Tool Extensions for Windows :

It adds the Generate Scripts wizard in the context menu of a selected database as in SSMS:

Please reference this tutorial: Generate data scripts using SSMS and Azure Data Studio .

We can not find any Extensions support script database as create in Azure Data Studio, it only supports script table as create with the Extension Simple Data Scripter:

For example:

Hope this helps.

Approach 1:

If python can be installed in the system, then we can use mssql-scripter to connect to the database and generate both schema and data scripts.

Installation: https://github.com/microsoft/mssql-scripter

Usage: https://github.com/microsoft/mssql-scripter/blob/dev/doc/usage_guide.md

Approach 2:

Since I couldn't get any new installations done in the remote system which I am working on and With the current extensions not supporting database scripting, I have to fall back to basic 'sys' schema to achieve this. Came up with a simple query to script the 'User Defined Table' definitions alone:

CREATE TABLE #CreateQueries
(
    QueryString NVARCHAR(MAX)
)

DECLARE @Counter INT 

DECLARE  
      @object_name SYSNAME  
    , @object_id INT  
    , @SQL NVARCHAR(MAX)  

SET @Counter=0
WHILE ( @Counter <= ( SELECT COUNT(*) FROM sys.objects WHERE type = 'U' ) )
BEGIN

    SELECT  
      @object_name = '[' + OBJECT_SCHEMA_NAME(o.[object_id]) + '].[' + OBJECT_NAME([object_id]) + ']'  
    , @object_id = [object_id]  
FROM (SELECT [object_id] = OBJECT_ID('dbo.' + name, 'U') FROM sys.objects WHERE type = 'U' ORDER BY name OFFSET @Counter ROWS FETCH FIRST 1 ROWS ONLY) o

SET @SQL = ''
      
SELECT @SQL = 'CREATE TABLE ' + @object_name + CHAR(13) + '(' + CHAR(13) +   
        
        TRIM(' ,' FROM (SELECT ' , [' + c.name + '] ' +   
        CASE WHEN c.is_computed = 1  
            THEN 'AS ' + OBJECT_DEFINITION(c.[object_id], c.column_id)  
            ELSE   
                CASE WHEN c.system_type_id != c.user_type_id   
                    THEN '[' + SCHEMA_NAME(tp.[schema_id]) + '].[' + tp.name + ']'   
                    ELSE '[' + UPPER(tp.name) + ']'   
                END  +   
                CASE   
                    WHEN tp.name IN ('varchar', 'char', 'varbinary', 'binary')  
                        THEN '(' + CASE WHEN c.max_length = -1   
                                        THEN 'MAX'   
                                        ELSE CAST(c.max_length AS VARCHAR(5))   
                                    END + ')'  
                    WHEN tp.name IN ('nvarchar', 'nchar')  
                        THEN '(' + CASE WHEN c.max_length = -1   
                                        THEN 'MAX'   
                                        ELSE CAST(c.max_length / 2 AS VARCHAR(5))   
                                    END + ')'  
                    WHEN tp.name IN ('datetime2', 'time2', 'datetimeoffset')   
                        THEN '(' + CAST(c.scale AS VARCHAR(5)) + ')'  
                    WHEN tp.name = 'decimal'  
                        THEN '(' + CAST(c.[precision] AS VARCHAR(5)) + ',' + CAST(c.scale AS VARCHAR(5)) + ')'  
                    ELSE ''  
                END +  
                CASE WHEN c.collation_name IS NOT NULL AND c.system_type_id = c.user_type_id   
                    THEN ' COLLATE ' + c.collation_name  
                    ELSE ''  
                END +  
                CASE WHEN c.is_nullable = 1   
                    THEN ' NULL'  
                    ELSE ' NOT NULL'  
                END +  
                CASE WHEN c.default_object_id != 0   
                    THEN ' CONSTRAINT [' + OBJECT_NAME(c.default_object_id) + ']' +   
                         ' DEFAULT ' + OBJECT_DEFINITION(c.default_object_id)  
                    ELSE ''  
                END +   
                CASE WHEN cc.[object_id] IS NOT NULL   
                    THEN ' CONSTRAINT [' + cc.name + '] CHECK ' + cc.[definition]  
                    ELSE ''  
                END +  
                CASE WHEN c.is_identity = 1   
                    THEN ' IDENTITY(' + CAST(IDENTITYPROPERTY(c.[object_id], 'SeedValue') AS VARCHAR(5)) + ',' +   
                                    CAST(IDENTITYPROPERTY(c.[object_id], 'IncrementValue') AS VARCHAR(5)) + ')'   
                    ELSE ''   
                END   
        END  
    FROM sys.columns c WITH(NOLOCK)  
    JOIN sys.types tp WITH(NOLOCK) ON c.user_type_id = tp.user_type_id  
    LEFT JOIN sys.check_constraints cc WITH(NOLOCK)   
         ON c.[object_id] = cc.parent_object_id   
        AND cc.parent_column_id = c.column_id  
    WHERE c.[object_id] = @object_id  
    ORDER BY c.column_id
    FOR XML PATH(''), TYPE)
    .value('.', 'NVARCHAR(MAX)') ) + 

    ISNULL( (   SELECT ', CONSTRAINT [' + i.name + '] PRIMARY KEY ' + i.type_desc + ' ' + 
                '[' + COL_NAME(ic.[object_id], ic.column_id) + ']' +  
                CASE WHEN ic.is_descending_key = 1  
                    THEN ' DESC'  
                    ELSE ''  
                END  
        FROM sys.index_columns ic WITH(NOLOCK)  
        JOIN sys.indexes i
        ON i.[object_id] = ic.[object_id]  
            AND i.index_id = ic.index_id 
            AND i.[object_id] = @object_id 
            AND i.is_primary_key = 1
            FOR XML PATH(''), TYPE)
    .value('.', 'NVARCHAR(MAX)'), '') + ');' + CHAR(13) +

    ISNULL( ( SELECT 'ALTER TABLE '+ @object_name + 
        ' ADD CONSTRAINT [' + f.name +'] FOREIGN KEY (['+
        COL_NAME(f_k_c.[parent_object_id], f_k_c.[parent_column_id])+ ']) REFERENCES '+ 
        '[' + OBJECT_SCHEMA_NAME(f_k_c.referenced_object_id) + '].[' + OBJECT_NAME(f_k_c.referenced_object_id) + ']' +
        ' (['+COL_NAME(f_k_c.[referenced_object_id], f_k_c.[referenced_column_id])+'])'+CHAR(13)+CHAR(10)+'GO'+CHAR(13)+CHAR(10)
        FROM sys.foreign_keys f WITH(NOLOCK)
        JOIN sys.foreign_key_columns f_k_c WITH(NOLOCK) ON
        f_k_c.constraint_object_id = f.object_id
        AND f.parent_object_id = @object_id
        FOR XML PATH(''), TYPE)
    .value('.', 'NVARCHAR(MAX)'), '')

    SET @Counter = @Counter + 1

    INSERT INTO #CreateQueries VALUES (@SQL)

  
END

SELECT * FROM #CreateQueries ORDER BY QueryString;

Hardcoded the 'dbo' schema for now. Can be extended further with other index scenarios as well to support as a good work around for the current scenario.