This hadeeth is fabricated. It was quoted by al-Shawkani (may Allah have mercy on him) in al-Fawaid al-Majmoo’ah fi’l-Ahadeeth al-Mawdoo’ah (p.44), as quoted in the question. He said: It was narrated by al-Jawzaqani from Anas (may Allah be pleased with him) in a marfoo’ report (i.e. attributed to the Prophet (peace and blessings of Allah be upon him)). But it is mawdoo’ (fabricated), and the men of its isnad (chain of narrators) are either majhool (unknown) or matrook (rejected).
Prayer at night is mustahabb (recommended) on all days, and is not restricted to what is mentioned in this fabricated hadeeth.
Allah says (interpretation of the meaning):
“Their sides forsake their beds, to invoke their Lord in fear and hope, and they spend (in charity in Allah’s Cause) out of what We have bestowed on them.
No person knows what is kept hidden for them of joy as a reward for what they used to do.”
“Verily, the Muttaqoon (the pious) will be in the midst of Gardens and Springs (in the Paradise),
Taking joy in the things which their Lord has given them. Verily, they were before this Muhsinoon (good-doers).
They used to sleep but little by night [invoking their Lord (Allah) and praying, with fear and hope].
And in the hours before dawn, they were (found) asking (Allah) for forgiveness.” [51:15-18]
Al-Tirmithi (3549) narrated from Abu Umamah (may Allah be pleased with him) that the Messenger of Allah (peace and blessings of Allah be upon him) said: “I urge you to pray qiyam al-layl (the optional night prayer), because that was the habit of the righteous who came before you; it will bring you closer to your Lord, expiate for bad deeds and keep you away from sin.”
Classed as hasan (sound) by al-Albani in Irwa al-Ghaleel, 452.
And Allah knows best.