The first Marlin lever gun came out in 1881, the first flat topped Marlin was made in 1889. The gun we know today was originally the model 94 marlin and was introduced in 1893 and later became known as the model 1936, shortened to 36 and today it's called the 336. Same basic design, same accuracy and same flat topped receiver suitable for mounting scopes.
As for the names, Winchester kept the model 94 designation, except for a few variations, and Marlin dropped the model 94 name after a bit. Which is one reason why when someone says model 94 you know they mean Winchester .30-30. My brother owned one in .32 special caliber, which was more or less the same as the .30-30 ballistic wise. Winchester also had name recognition on their side due to the fact they were making rifles before Marlin started up.
As for who made the first lever gun, Winchester or Marlin, that would be Winchester, or to be exact B. Tyler Henry with the Henry repeater which quickly morphed into the 1866 model Winchester after Oliver Winchester acquired the factory. However, Marlin brought out their version of the 94 in .30-30(and various other calibers including .38-55)before the venerable Model 94, albeit only about a year sooner.
PS Marlin still makes the model 1894, they brought it out again about 1979 or so. Today it is one of the most popular models for cowboy action shooting.