A noun in The Etymologies of the 1930s glossed “small bird” derived from the root ᴹ√PHILIK and cognate to ᴹQ. filit (filik-) of the same meaning (Ety/PHILIK). Tolkien said fileg was an “analogical singular” form (along with another singular form filigod) based on its plural form filig. That is because the final -k in ✱philik was lost in ancient times, as explained in Common Eldarin: Noun Structure from the early 1950s (PE21/72) so that its historical phonetic development would have produced singular N./S. ✱fil. This form was deemed unsuitable, and a new singular form fileg was constructed based on the plural. Something similar happened with S. thoron “eagle”.
Conceptual Development: The Gnomish Lexicon of the 1910s had G. bilin or bilinc for “a small bird, especially sparrow”, a diminutive form of G. bil “bird” (GL/22, 23).
Neo-Sindarin: I think it is likely that fileg can also be used to refer to sparrows for purposes of Neo-Sindarin, since its Quenya cognate filit was in one place glossed “sparrow, small bird” (PE21/56).
This root first appeared as ᴹ√PHILIK “small bird” in The Etymologies of the 1930s with derivatives ᴹQ. filit and N. fileg of the same meaning (Ety/PHILIK). The root also appeared in the contemporaneous Primitive Quendian Structure: Final Consonants with an extra gloss “sparrow”, a deleted variant spilik-, and some additional (unglossed) derivatives ᴹQ. filinke and N. flinc (PE21/56). The root and these last two derivatives reappeared in Common Eldarin: Noun Structure from the early 1950s, in this document with the gloss “finch” (PE21/71-72, 80-81). Note that a likely early precursor to all these words was G. bilinc “sparrow, bird (small)” (GL/23).
Neo-Eldarin: For purposes of Neo-Eldarin, I think it is best to assume the root √PHILIK = “small bird”, with ᴹQ. filit/N. fileg = “sparrow” and Q. filincë/S. flinc = “finch”.