Here is the relevant code:

        $params = @{
            Query = $query
            ServerInstance = $serverInstance
            Database = $database
            Verbose = $true
            OutputSqlErrors = $true
            AbortOnError = $true
            IncludeSqlUserErrors = $true
        }

        try{
            $result = Invoke-Sqlcmd @params
            return $result
        }catch{
            Write-Host $_.Exception.Message  # Display the error message
            Write-Host $_
            throw $_.Exception.Message
        }

When I run this on the command line (PS shell), I get ONLY the following:

Incorrect syntax was encountered while parsing ''.
Incorrect syntax was encountered while parsing ''.
At [].psm1:56 char:13
+             throw $_.Exception.Message
+             ~~~~~~~~~~~~~~~~~~~~~~~~~~
    + CategoryInfo          : OperationStopped: (Incorrect synta...ile parsing ''.:String) [], RuntimeException
    + FullyQualifiedErrorId : Incorrect syntax was encountered while parsing ''.

However, when I run the exact same code in my Azure pipeline, the logs there show the full message:

Cannot open database "[dbName]" requested by the login. The login failed.
Incorrect syntax was encountered while parsing ''.
Login failed for user '[username]'.Incorrect syntax was encountered while parsing ''.
At [].psm1:55 char:13
+             throw $_.Exception.Message
+             ~~~~~~~~~~~~~~~~~~~~~~~~~~
+ CategoryInfo          : OperationStopped: (Incorrect synta...ile parsing ''.:String) [], RuntimeException
+ FullyQualifiedErrorId : Incorrect syntax was encountered while parsing ''.

How do I get the full output (the entire error message) on the command line (PS shell)?

**Update: With help from @mklement0 I've updated my code to

        }catch{
            Write-Host $_  # Display the error message
            $_ | Format-List * -Force | Out-Host
            throw
        }

this gives me the stack trace where the exception is Microsoft.SqlTools.ServiceLayer.BatchParser.BatchParserException

To me, this suggests that maybe the parser is getting the original response/error message from the query, trying to parse it, and then throwing its own exception, instead of relaying the original one. Is there anything to do about this?