Yes, there are different usages here.
'... a tiny bit sad ...' and 'that tiny bit better' (compare 'somewhat') show adjective-premodification (here, downtoning: contrast intensifying, with 'very/extremely'; 'much'), and are still labelled in this role as (compound, ie multi-word) 'adverbs' by some traditionalists. I'd go with the 'splitting rather than lumping' analysis explained by Hurford James R in 'Grammar: A Student's Guide' (p9) here:
Some words, such as very, quite, rather, and somewhat, which
modify adjectives and adverbs, are sometimes [themselves also] called
adverbs, but that is not a very good name for these, because all sorts
of other words are also called adverbs, and the term begins to get
vague and unspecific. (Actually, linguists use the term ... degree
modifier [secondary modifier; intensifier / downtoner] for very,
quite, rather, and somewhat [and awfully, alarmingly, incredibly, seriously ... when used to modify adjectives/adverbs].
The term 'degree modifier' is well known, but 'secondary modifier (of adjective, etc)' is more fitting for the whole class, as 'alarmingly', 'strappingly', 'calculatedly ... carry semantic weight far beyond the mere intensification / downtoning words live 'very', 'slightly' carry.
'Slightly sad' / 'a bit sad' show downtoner usage; 'a tiny bit sad' shows a mitigated downtoner (second order, with the downtoner itself being downtoned!) usage.
On an even keel, a tiny bit sad, a bit sad, rather sad, sad, very sad, worryingly sad, extremely sad....
'[A] tiny bit of' is a [compound] [intensified; the archetypal compound quantifier is 'a bit of'. Note that here, 'tiny' reinforces 'bit' (downwards!), rather than mitigating an adjective as in 'a tiny bit better'] quantifier.
As explained in an article on Grammar put out by the British Council:
Quantifiers with uncount nouns
Some quantifiers can be used only with uncount nouns:
(not) much // a bit of // a little
Would you like a little wine?
Could I have a bit of butter, please?
I've avoided the word 'idiom', as premodification of adjectives is totally standard grammar, however it is analysed. No words are used with unusual senses either (as @Jason Bassford et al points out), so 'idiom' is not applicable (and I'd use 'compound premodifier' and 'compound quantifier' rather than 'fixed expression' here myself). But I understand why you're asking about the wisdom of treating say 'a bit of' as a single lexeme. Analysis into say [indefinite article] + [noun] + [preposition] doesn't get us very far, when 'a little' and 'a bit of' are largely interchangeable.