Below is the OED entry to "livelong." Obviously, in the show title and songs sense 1 ("very long; whole, entire") is involved; and yet the sense of living kicks in there whenever you use "livelong." As far as I understand these things, yes, the β. forms do seem to stem from the same root as G. "lieb," if that's what you meant.
α. late Middle English– livelong, 1500s–1600s liuelong, 1700s liblong.
β. late Middle English lefelong, late Middle English lefelonge, late Middle English levelonge, 1500s leeuelong, 1700s– leelang (Scottish), 1800s lee-long (Irish English (northern)).
Origin: Formed within English, by compounding. Etymons: lief adj., long adj.1, live v.1
Etymology: Originally < lief adj. + long adj.1 (as reflected by β. forms). Compare Middle Dutch (rare) den lieven langhen nacht , lit. ‘the dear long night’, German den lieben langen Tag (1582 or earlier), die liebe lange Nacht (1670 or earlier; 1609 or earlier as ein liebe lange nacht ), das liebe lange Jahr (17th cent. or earlier), lit. ‘the dear long day’, ‘the dear long night’, ‘the dear long year’. The α. forms show alteration of the first element after live v.1 (forms of which with stem vowel -e- probably facilitated the reanalysis). Compare all the long day, all the long night (see long adj.1 and n.1 Phrases 3b). Compare later live adj.2
Sense 2 may well show an independent formation, < live v.1 + long adj.1, perhaps partly suggested by earlier currency in sense 1.
In sense 3 perhaps another independent word, if so probably a variant of lifelong adj. with voiced consonant, perhaps by association with livelong adj. in either sense 1 or sense 2. N.E.D. (1903) notes at sense 3 ‘Probably meant to be pronounced (ləiv) /laɪv/’, but there is no clear evidence for this.
poetic and rhetorical.
1. As an intensified form of long. Of a period of time: very long or apparently very long; whole, entire. Chiefly in the livelong day (also night). See also live adj.2
2. That lives long, that endures; lasting.
3. Lasting or continuing for a lifetime; = lifelong adj.