Yes galaxie is right, the reasoning is that marriage must be lawful out of respect to the authorities—Romans 13. That’s why JWs must have marriages legally recognised.
Oppostate is also correct, because Watchtower were not always as respectful of authorities (they denied that Romans 13 applied to governments from 1929 to 1962) and for a period they actually discouraged couples from getting their marriages legally recognised. Does that mean some JW couples lived together without getting married in that period? It seems to. Which sounds pretty bizarre compared with current JW attitudes. But it was only for a brief period in the 1940s and 50s I believe.