A dropdown for something with that many options would make for really bad UX and lead to a far too much scrolling if the year you're interested in isn't immediately shown.
A better idea imo is to dynamically change the minimum and maximum year values in the current slider to match the oldest and newest anime/manga that each user has on their list. After all, comparatively few users actually have entries in their list that go back as far as 1951.
On mobile, the slider is virtually impossible to use. It basically became a game of 'move the slider far to the left and just randomly press and hope it gives you the year you wanted'. Via mouse on a desktop it's still difficult to use as many times it goes one to far or one to few.
I think a more ideal solution would be a slider and then the display can be manually edited. So a two-way binding between a slider and a text box.