Are NATO fly-by-wire fighters designed to feel like F-14 Tomcats?
Most decidedly not, at least not for the first fly-by-wire fighter in the NATO arsenal, the F-16. Its stick, in initial block versions, did not move at all, instead using piezoelectric load cells to detect stick pressure. This caused some issues with novice Viper drivers overreacting to the stiffness of the controls compared to anything they had flown and overrotating the jet on takeoff, smacking the turkey feathers on the ground. Later blocks put a little play in the stick.

As far as the Tomcat being an easy plane to fly, Top Gun kind of over-romanticized it. The TF30 engine initially put into these aircraft was underpowered and terribly unreliable (one thing the movie did get right is that the engines were prone to compressor stalls when encountering disturbed air such as jetwash or even their own missile launch trails). The aircraft itself, while a big improvement in dogfighting over the F-4, was still big and heavy, the result of requirements that it be a long-range supersonic interceptor in an era without supercruise or the relatively light AMRAAM missile; the aircraft instead had to be able to haul a full load of AIM-54 Phoenix missiles intended for the ill-fated naval variant of the F-111, along with the large, complex radar system to lock on and fire them, and a boatload of fuel to get all that out to patrol range and back. 

The Navy's initial mixed results with the F-14A, along with the extremely high costs of the aircraft ($32 million each at the height of production) and the lack of ground attack capabilities, were mirrored in the USAF with its F-15 program; the looming retirement of the F-4 was about to leave both branches without a fighter capable of the Phantom's multi-mission capability, and both the F-14 and F-15 were big expensive interceptor-style pure air-to-air fighters, which some Air Force brass believed was proof that nothing had been learned from Vietnam. The USAF commissioned the Light Weight Fighter competition, which produced the F-16, a light, relatively inexpensive multirole fighter that has become one of the most successful land-based fighter jets in history. The Navy preferred some traits in the USAF's losing aircraft, the YF-17, such as the twin-engine design for durability, and it would go on to develop this design into one of the most successful naval aircraft of all time, the F/A-18 Hornet and Super Hornet. 

The Tomcat eventually lived up to its promises with some engine and avionics upgrades (including a hybrid-FBW system and a LANTIRN pod to add air to ground smart bomb capabilities) but the aircraft was by no means a perfect plane or a dream to fly.