We can find examples of this as early as the early 20th century with jazz bands. The jazz and swing scenes seems to hold the most promising early examples for this - with band names such as 'The Original Dixieland Jass Band' (1917) 'The Ramblers' (a Dutch band from 1926), 'Red Hot Peppers' (also 1926) and Harlem Hamfats (1936). You also get a lot of animal descriptors for groups in older blues/folk from the late 1918 period onwards, like 'Bullmoose Jackson & His Buffalo Bearcats', but I wouldn't strictly say that's a bandname since it includes a literal description of the lead singer - 'Benjamin Clarence "Bull Moose" Jackson'.

But if you want band names that don't start with 'The', then you need to jump to the 60s when we get bands such as Them (1963/1964), Cream (mid 1960s), Our Lost Souls (1966). Modern Jazz Quartet were known in 1952 to do this, but it is arguable that since some people used a 'the' in their name, they don't count.