Most likely Mouryou Senki Madara or simply Madara, a 1991 two-part movie.
Born of legendary origins, the young Madara has grown up in obscurity as an apprentice blacksmith. An encounter with the demonic Mouki reveals his destiny, and catapults him into the middle of a mystic quest. Madara fights for revenge with the help of freedom fighters, magical swords and Kirin, the master of the Majin. His goal is to defeat his brother and then overthrow his father, the despotic Miroku.
Both parts are available on YouTube; relevant timestamps from the first part:
- You can see Madara fighting a bat-headed demon at 26' onwards; Madara's "gimmick" arm is seen in action at 27'43 (mechanical arm extends);
- 28'42, Madara's origin story is revealed. He's the son of Miroku, a "demonic overlord" and Princess Sakuya of the Ugara royal lineage.
And timestamps from the second part:
- around 15'00, the brother talks to his (missing) mother while looking at a rocking chair;
- explanation about the swords combining start at 43'28. Actual merging is seen at ~49'10.
There's a legend in the Ugaya Royal Family, when the two Kusanagi become one, a true ruler shall appear on Earth and bring forth peace. But when the two Kusanagi become one, it's after one of the owners' blood is shed!
Found with the Google query
anime arm "torn off * demon" "mechanical arm" which returned this very question and a repost on MyAnimeList forums. Someone IDed it there.