# I will clothe him with your tunic and put on him your sash

Yahweh causing Eliakim to take Shebna's place in the king's palace is spoken of as if Yahweh will dress Eliakim in Shebna's clothes that represent his authority in the king's palace. 

# sash

This is a piece of cloth that people wear around the waist or across the chest.

# into his hand

"to him" 

# He will be a father

"He will be like a father" 

# to the house of Judah

"to the people of Judah"