starched is used figuratively:
OED
starch (v.) 3.a. transitive. To make (oneself, a person, the face or features, etc.) formal, severe, or haughty; to make
(discourse, speech, etc.) formal or excessively serious.
2009 S. Walker Power of Tolkien's Prose v. 143 The dwarvish
dialect is formal and formulaic, ‘a fair jaw-cracker’ starched with
ceremonial phrase.
From which
Starched: 2. figurative. Stiff, formal; prim; = starchy adj.
a. Of a person, a person's countenance, behaviour, etc.
1837 C. Dickens Pickwick Papers xxvii. 276 His looks were
starched, but his white neckerchief was not.
2006 Psychologies (U.K. ed.) July 45/4 She has set her heart on a handsome but heavily starched academic..who is poised to be unhappily married to a frumpy killjoy.
I have heard "he is a starched collar" (where "collar" is synecdoche - compare "a suit from head office")) in the sense of a stiff/formal/prim person but can find no examples as "starched collar" is so common in its literary meaning.