So far, as far as I can tell, we have --
different reported birth dates
different reported names (Frank? Patrick?)
unknown wherabouts
Possible desertioin by father with a subsequent bigamous marriage (but if this was the right Thomas, then Frank/Patrick was conceived whilst he supposedly "married" to Alice Vickery so that could open a whole new can of worms)
different dates and places for attestation into the army
With all that, I wouldn't rule out 10 yr old Frank Reen as a Barnardo's boy.
Just to add to the fun, there are various court reports naming Ellen Vary -- don't know if it's the same Ellen, but one in 1893 for allegedly stabbing her husband, and many over the years, mostly in the south east of England, for being drunk and disorderly.
In Oct 1902, one Ellen Vary was begging in New Bond St and George St - this was reported in the Bath Chronicle and I get the impression these were local addresses, not London ones. She appeared in court with a baby in her arms, saying her husband had deserted her.