Easyjet staff get some perks, including a good staff travel scheme on their own flights. Unfortunately, unlike many other carriers, they choose not to offer anything to the staff who work for their handling agents.
Ground handling staff don't get a choice in the uniform they wear - if a particular contract includes wearing the airline uniform, then thats what you have to wear. And, no, the majority don't tend to think they work for the carrier in question! When passengers are complaining or things are going wrong, staff are very quick to flash their airport id and point out they do NOT actually work for the airline!!!
With regards to dedicated uniformed staff - when a contract changes hands, TUPE usually applies and staff will have the option of a transfer to the new handling agent!