Joining subquery from same table on 1=1 in Oracle
I think it should be this:

SELECT 
    zip.zip,
    ROUND(SDO_GEOM.SDO_DISTANCE((MDSYS.SDO_GEOMETRY(2001, 4326, MDSYS.SDO_POINT_TYPE(inp.lng, inp.lat, NULL), NULL, NULL)), (SELECT geom FROM standard_zip WHERE zip = zip.zip), 0.005) / 1609.34, 2) AS dist,
    inp.lat,
    inp.lng
FROM standard_zip zip
   CROSS JOIN (SELECT x as lng, y as lat from standard_zip where zip = :zip) inp
WHERE zip.zip IN (
        SELECT zip
        FROM standard_zip c
        WHERE SDO_FILTER(c.geom, SDO_UTIL.CIRCLE_POLYGON(inp.lng, inp.lat, (:r * 1609.34), 5)) = 'TRUE'
            AND SDO_RELATE(c.geom, SDO_UTIL.CIRCLE_POLYGON(inp.lng, inp.lat, (:r * 1609.34), 5), 'mask=ANYINTERACT querytype = WINDOW') = 'TRUE'
        )
ORDER BY DIST;