In his character design for Final Fantasy VII, Cloud bears heavy metal armor on his left shoulder and wears a heavy bracer on his left arm. The bracer and pauldron are both roughly machined and covered in rivets, implying their use as a restraining device rather than as protective armor.
And in the Final Fantasy VII: Advent Children film, Cloud does gain a reason to “seal away” his left arm: Geostigma.
According to the post, early concepts for the protagonist of FFVII characterized Cloud (or rather, the character that would eventually be Cloud Strife) as a “Berserker.”
An uncontrollable warrior, this berserker version of Cloud would have their left arm “sealed away” by his bracelet.