The effect of asking someone to supplicate for you on your sincerity

Is the statement a person makes to his brother, “Supplicate for me!” a mark against his sincerity and reliance upon Allaah?

No, this does not affect it. If someone sees from his brother’s righteousness which makes him believe that their supplication will be answered, then he can request from them to supplicate for him, and it does not tarnish his reliance upon Allaah. In a narration reported by Muslim, the Prophet ﷺ ordered ‘Umar the caliph of the Muslims that when he sees Uwais Al-Qarny, he should request him to ask Allaah for forgiveness for him, and that was someone lesser in status than him. Therefore, this request does not affect a person’s reliance. There is another narration from the Prophet ﷺ in which he said to ‘Umar,

“Do not forget us when you make supplications, my brother!” (Abu Daawood 1498)