>By any chance is there a FireWall on the W98 Machine?

No. Only AVG antivirus is present. I turned AVG resident shield off but the problem persists.
Also I turned Postgres ODBC driver logging ON. Driver log shows that driver seems to return OK. So that error does not in driver code.

[-1578905]globals.extra_systable_prefixes = 'dd_;'
[-1578905][[SQLAllocHandle]][-1578905]**** in PGAPI_AllocEnv **
[-1578905]** exit PGAPI_AllocEnv: phenv = 28970736 **
[-1578905][[SQLSetEnvAttr]] att=200,2
[-1578905][[SQLAllocHandle]][-1578905]PGAPI_AllocConnect: entering...
[-1578905]**** PGAPI_AllocConnect: henv = 28970736, conn = 27869760
[-1578905]EN_add_connection: self = 28970736, conn = 27869760
[-1578905] added at i =0, conn->henv = 28970736, conns[i]->henv = 28970736
[-1578905][SQLGetInfoW(30)][-1578905]PGAPI_GetInfo: entering...fInfoType=77
[-1578905]PGAPI_GetInfo: p='03.00', len=10, value=0, cbMax=12
[-1578905][SQLSetConnectAttrW][-1578905]PGAPI_SetConnectAttr 102
[-1578905]PGAPI_SetConnectOption: entering fOption = 102 vParam = 1
[-1578905][SQLSetConnectAttrW][-1578905]PGAPI_SetConnectAttr 103
[-1578905]PGAPI_SetConnectOption: entering fOption = 103 vParam = 15
[-1578905][SQLSetConnectAttrW][-1578905]PGAPI_SetConnectAttr 112
[-1578905]PGAPI_SetConnectOption: entering fOption = 112 vParam = 4096
[-1578905][SQLDriverConnectW][-1578905]PGAPI_DriverConnect: entering...
[-1578905]**** PGAPI_DriverConnect: fDriverCompletion=1, connStrIn='DRIVER={PostgreSQL Unicode};UID=eetasoft;DATABASE=eeva;SERVER=1.2.3.4;PORT=5432;B9=0;'
[-1578905]our_connect_string = 'DRIVER={PostgreSQL Unicode};UID=eetasoft;DATABASE=eeva;SERVER=1.2.3.4;PORT=5432;B9=0;'
[-1578905]attribute = 'DRIVER', value = '{PostgreSQL Unicode}'
[-1578905]copyAttributes: DSN='',server='',dbase='',user='',passwd='xxxxx',port='',sslmode='',onlyread='',conn_settings='',disallow_premature=-1)
[-1578905]attribute = 'UID', value = 'eetasoft'
[-1578905]copyAttributes: DSN='',server='',dbase='',user='eetasoft',passwd='xxxxx',port='',sslmode='',onlyread='',conn_settings='',disallow_premature=-1)
[-1578905]attribute = 'DATABASE', value = 'eeva'
[-1578905]copyAttributes: DSN='',server='',dbase='eeva',user='eetasoft',passwd='xxxxx',port='',sslmode='',onlyread='',conn_settings='',disallow_premature=-1)
[-1578905]attribute = 'SERVER', value = '1.2.3.4'
[-1578905]copyAttributes: DSN='',server='1.2.3.4',dbase='eeva',user='eetasoft',passwd='xxxxx',port='',sslmode='',onlyread='',conn_settings='',disallow_premature=-1)
[-1578905]attribute = 'PORT', value = '5432'
[-1578905]copyAttributes: DSN='',server='1.2.3.4',dbase='eeva',user='eetasoft',passwd='xxxxx',port='5432',sslmode='',onlyread='',conn_settings='',disallow_premature=-1)
[-1578905]attribute = 'B9', value = '0'
[-1578905]copyAttributes: DSN='',server='1.2.3.4',dbase='eeva',user='eetasoft',passwd='xxxxx',port='5432',sslmode='',onlyread='',conn_settings='',disallow_premature=-1)
[-1578905]our_connect_string = 'DRIVER={PostgreSQL Unicode};UID=eetasoft;DATABASE=eeva;SERVER=1.2.3.4;PORT=5432;B9=0;'
[-1578905]attribute = 'DRIVER', value = '{PostgreSQL Unicode}'
[-1578905]CopyCommonAttributes: A7=100;A8=8192;A9=0;B0=254;B1=8190;B2=1;B3=1;B4=1;B5=1;B6=0;B7=1;B8=0;B9=1;C0=0;C1=0;C2=dd_;[-1578905]attribute = 'UID', value = 'eetasoft'
[-1578905]CopyCommonAttributes: A7=100;A8=8192;A9=0;B0=254;B1=8190;B2=1;B3=1;B4=1;B5=1;B6=0;B7=1;B8=0;B9=1;C0=0;C1=0;C2=dd_;[-1578905]attribute = 'DATABASE', value = 'eeva'
[-1578905]CopyCommonAttributes: A7=100;A8=8192;A9=0;B0=254;B1=8190;B2=1;B3=1;B4=1;B5=1;B6=0;B7=1;B8=0;B9=1;C0=0;C1=0;C2=dd_;[-1578905]attribute = 'SERVER', value = '1.2.3.4'
[-1578905]CopyCommonAttributes: A7=100;A8=8192;A9=0;B0=254;B1=8190;B2=1;B3=1;B4=1;B5=1;B6=0;B7=1;B8=0;B9=1;C0=0;C1=0;C2=dd_;[-1578905]attribute = 'PORT', value = '5432'
[-1578905]CopyCommonAttributes: A7=100;A8=8192;A9=0;B0=254;B1=8190;B2=1;B3=1;B4=1;B5=1;B6=0;B7=1;B8=0;B9=1;C0=0;C1=0;C2=dd_;[-1578905]attribute = 'B9', value = '0'
[-1578905]CopyCommonAttributes: A7=100;A8=8192;A9=0;B0=254;B1=8190;B2=1;B3=1;B4=1;B5=1;B6=0;B7=1;B8=0;B9=0;C0=0;C1=0;C2=dd_;[-1578905]CC_connect: entering...
[-1578905]CC_connect(): DSN = '', server = '1.2.3.4', port = '5432', sslmode = 'prefer', database = 'eeva', username = 'eetasoft', password='xxxxx'
[-1578905]connecting to the server
[-1578905]connecting to the database using 1.2.3.4 as the server
[-1578905]the size is 19
[-1578905]Could not establish connection to the database; LIBPQ returned -> fe_sendauth: no password supplied

[-1578905]CONN ERROR: func=PGAPI_DriverConnect, desc='Error from CC_Connect', errnum=113, sqlstate=08001, errmsg='fe_sendauth: no password supplied
'
[-1578905][SQLGetDiagRecW][-1578905]PGAPI_GetDiagRec entering rec=1[-1578905]**** PGAPI_ConnectError: hdbc=27869760 <0>
[-1578905]enter CC_get_error
[-1578905]enter CC_create_errormsg
[-1578905]msg = 'fe_sendauth: no password supplied
'
[-1578905]exit CC_create_errormsg
[-1578905]exit CC_get_error
[-1578905]CC_get_error: status = 113, msg = #fe_sendauth: no password supplied
#
[-1578905] szSqlState = '08001',len=34, szError='(null)'
[-1578905]PGAPI_GetDiagRec exiting 1
[-1578905][SQLGetDiagRecW][-1578905]PGAPI_GetDiagRec entering rec=1[-1578905]**** PGAPI_ConnectError: hdbc=27869760 <36>
[-1578905]enter CC_get_error
[-1578905]exit CC_get_error
[-1578905]CC_get_error: status = 113, msg = #fe_sendauth: no password supplied
#
[-1578905] szSqlState = '(null)',len=34, szError='fe_sendauth: no password supplied
'
[-1578905]PGAPI_GetDiagRec exiting 0
[-1578905][[SQLGetDiagFieldW]] Handle=(2,1a94240) Rec=1 Id=8
[-1578905]PGAPI_GetDiagField entering rec=1, HandleType=2, DiagIdentifier=8
[-1578905]PGAPI_GetDiagField exiting 0