i started classic strategy games by go. maybe it helped me for shogi, but at very long term. i barely knew how to play chess really, and so started there s 10 years from scratch. it s evident that there s a relation between go and shogi, the drop rule, but the fact that in go all pieces have the same caracteristics change all. but certainly go and shogi teach a way of acting, spiritualy that can be compared. patience, awareness, calm...
I can't speak for Go, but I can say from first-hand experience that yes, unequivocally, other chess variants help in shogi.
I played chess competitively in high school. I picked up shogi a few years later. Having a strategy background helps learn the game quickly, but more importantly, having the patience to sit and study the board is an important part in both games.
There are still a lot of differences between the games, but the most important aspect is knowing how to study and improve.
I've seen similar patterns in our club's new members. Those with a strong chess background are able to adopt shogi much more quickly and become strong players without studying extensively.
I imagine that Go helps in shogi and vice versa, but shogi variants like chess and xiangqi are probably better due to their shared ancestry.