First, background info. A visual approach can have various meanings. In the case of airlines it does not convert an IFR flight to VFR nor cancel the flight plan. What ATC is doing [in the USA] is asking a pilot "do have the runway in sight?" or "traffic 4 miles ahead and 1000 feet below, B737, do you have it in sight?", if the pilot says no than ATC maintains positive control and minimum radar separation, if the pilot says yes then ATC can issue a visual approach either directly if the runway is in sight or with "follow that plane, maintain visual separation, cleared to land runway 35R" even if the runway is not yet visible. As the pilot is maintaining separation (with a specific aircraft only) the radar minima are no longer in effect and reduces ATC workload so they can focus on other problems like looking for aircraft that are off course, and it allows much closer spacing and landing. Tower based visual separation is often used for departures before radar minima are achieved, this is a controller in the tower looking at both planes out of the window.
Now why do American pilots favor visual separation and clearance?
A cultural difference, in part due to the supply of pilots and general aviation, in part due to the general culture of the country. The majority of general aviation is in north America and so US airline pilots are often sourced from general aviation pilots that are accustomed to visual flight procedures. The USA also has traditionally had a very large supply of military pilots, who are even more accustomed to adapting to non-standard conditions and visual separation.(like in flight refueling) Often military convoys are simply given block altitude corridors by ATC and the pilots fly as they like within that block separating themselves.
With FAA regulations, while under positive ATC radar separation both the pilot and controller are more at risk of a regulation violation if minimum separation is not maintained. There is no explicit minimum for two planes under visual separation.(radar separation still applies to other traffic)
Airline companies outside of the USA, due to a lack of general aviation and smaller military, tend to train pilots "ab-initio". That is complete training from the beginning, often targeted at a specific plane and airline procedures. This is more narrowly focused than typical American training; this is efficient but some of the programs may lack breadth. I tend to think this focus can make a very good co pilot in less time but maybe is not as good for captains.(only an opinion and not a strong opinion) Funny enough many of these airlines have their training programs in the USA for reduced cost and may use US instructors, but it is not typical US curriculum.
Now for company culture, non-USA airlines often encourage full use of automation to avoid pilot deviation from standards. This includes using radar separation rather than visual, computer limited flight controls(Airbus "normal law"), and heavy use of autopilot even at lower altitudes. This has the side effect of more pilots loosing the ability to handle alternate situations. So you have reduced problems created by pilot judgment but may also reduce pilot judgment when there is a problem. Air France 447 is the most recent example with a full investigation report.(LionAir610 may also be an example based on the preliminary report, but the investigation is still in progress.) I do not know the legal responsibility in each country, maybe airline company, maybe pilot, etc.
On the US side, the pilot in command is legally responsible for the whole flight from from preflight log review and general aircraft inspection all the way to engine shutdown and unloading. As such the pilot has final authority to accept or refuse any ATC instruction. Though this has some limits; if you say yes to do something and then do something different this is a "pilot deviation" and may require paperwork to explain. US ATC often asks pilots to "state intentions" if the pilot contacts them with a problem, that is the pilot tells ATC what the pilot is going to do rather than asking for permission. Basically you may hear pilots outside the USA asking permission to maneuver or land, even in an emergency; in the USA pilots are trained to take any action needed to get the plane on a runway in one piece and to inform ATC is second priority(helpful but not essential).