The earliest recorded use of dicky-bird in this childlike way was from Horace Walpole in 1791, so it has been with us for a long time. It's rather like the children's word 'ickle' for 'little' and similar examples. Adults simply make these up in an attempt to create baby language; other than that, there is no specific reason for such inventions.
Puss is older than dicky-bird, having been around since the early 16th century. It almost certainly started life simply as the 'pss-pss' sound we naturally make when we actually try to call a cat to us.