There is an Albert Barr born in Q4 of 1889 in Birmingham on ancestry .... the only Albert James Barr was born in 1886 in Nottingham, so as you mention Birmingham, I should say he is the first one .. it's not uncommon for middle names to change or be missing or added later......... And this same Albert Barr shows on the 1891 census as born about 1890 (if he was born in Dec 1889 he may not have been registered until Jan 1890) he is with his parents Emma L and Stewart R and several siblings at number 6 House, 56 Hatchett Street, St Stephens, Birmingham. The address would suggest it's a "court" or back to back house, and Hatchett Street still exists (just off A4540 in the Summerhill area of Birmingham)
On the 1901 census he is 11 and living with same named parents in Lozells, Birmingham, again with several siblings, his father is still a gun action filer and was born in Woolwich (listed then as Kent) and their address is 14 court, Guildford Street, Aston Manor, Lozells.....
Hope this helps
EK