Partition list at repeated element

Join[Most /@ Rest @ Most @ #, {Last @ #}] & @ Split[list, UnsameQ]

{{a, 1, 2, 3}, {b, 5, 6}, {c, 1, 5}, {a, 7, 8, 9, 1}}

You can also use use Split twice and reorganize the result:

Rest /@ Flatten /@ Split[Split[list], Length @ #2 == 1 &]

{{a, 1, 2, 3}, {b, 5, 6}, {c, 1, 5}, {a, 7, 8, 9, 1}}

And yet an alternative way:

Take[list, {#, #2 - 2}] & @@@ 
 Partition[Last /@ SequencePosition[list, {a_, a_}], 2, 1, 1, {Length[list] + 2}] 

{{a, 1, 2, 3}, {b, 5, 6}, {c, 1, 5}, {a, 7, 8, 9, 1}}

SequenceSplit[list, {a_, a_, x:_Integer ..} -> {a,x}]

{{a, 1, 2, 3}, {b, 5, 6}, {c, 1, 5}, {a, 7, 8, 9, 1}}