mate, you are not 'transferring', you are finishing 1 flight, and then checking in for another.
as for who holds the tickets, who have you booked your flights through? both EZY & FR do not do 'tickets' anymore, in fact all you need is your reference number or (if you prefer) a copy of your confirmation.
on another point, i sincerely hope that the EZY part of the flight is nor delayed, as rest assured the FR flight will not wait for you. after all, why should Mr oleary have to delay his plane just because his competitor Mr Stelios cannot get his flight to depart on time.
IMHO, i would have left a LOT MORE TIME in between!