MS Access is many, many things all jammed together into the same box.
MySQL is a relational database.
It's not going to go fancy "look-up" stuff for you because that's not its job. MySQL will store you data (very well) but you're going to have to use
something else to build the User interface; PHP is well favoured around here.
Also, I'd suggest using an autoincrement key for your Movies table - you can easily get two movies with the same name and you don't want them getting mixed up.
(Warning: Air-code; probably contaminated by "full-blown" Oracle-thinking)
create table movie_list
( ID integer not null auto_increment
, Title varchar( 250 ) not null
, primary key ( ID )
) ;
create table user_list
( ID integer not null auto_increment
, Name varchar( 200 ) not null
, primary key ( ID )
) ;
create table watch_lists
( User_ID integer not null
, Movie_ID integer not null
, primary key ( User_ID, Movie_ID )
, foreign key User_ID references user_list ( ID )
, foreign key Movie_ID references movie_list ( ID )
) ;
insert into movie_list( Title )
values ( 'Jaws' ), ( 'Jaws 2' ), ( 'Jaws 3' ), ( 'Bambi' ) ;
insert into user_list( Name )
values ( 'Fred' ), ( 'Wilma' ), ( 'Barney' ), ( 'Betty' ) ;
insert into watch_lists
values ( 1, 4 ), ( 2, 1 ), ( 3, 1 ), ( 4, 2 ) ;
select u.name
, m.Title
from watch_lists w
inner join user_list u
on w.user_id = u.id
inner join movie_list m
on w.movie_id = m.id
order by u.id, m.id ;
+--------+--------+
| Name | Title |
+--------+--------+
| Fred | Bambi |
| Wilma | Jaws |
| Barney | Jaws |
| Betty | Jaws 2 |
+--------+--------+
Regards, Phill W.