Main question: they come from the same PIE root.
The verb fīō comes from (the zero-grade of) the PIE root *bhuH-, "to become". This led to a Proto-Italic verb *fui-. In Proto-Italic, the phoneme *f is thought to have been a bilabial fricative, voiceless [ɸ] at the start of a word and voiced [β] everywhere else.
Somewhere around Proto-Italic, this verb *fui- started to be used for certain periphrastic formations: *fuā- (subjunctive) for the imperfect, and *fui- (indicative) for the future. Then, just as in the verb fīō itself, the *u disappeared before these other vowels.
This left Proto-Italic (or Post-Proto-Italic) forms like *amā-fā-s "you were loving" and *amā-fi-s "you will love". In other Italic languages, this medial *f remained: Oscan fufans "they were" (= Latin *fubant = erant), Faliscan carefo (= Latin carēbō). But in Latin, medial *f [β] hardened into a stop b, giving the endings -bās and -bis and separating them thoroughly from the verb fīō.
See this paper for more information. This paper also points out that these tenses are also seen in Celtic, and that their origin is much less cut-and-dry than I make it sound.