President of the United States of America.
For the past centuries, many have been me. Many more desire to become me. Some are forgotten, others are remembered.
In all, since the 18th century there have been 45 US Presidents (including Grover Cleveland's non-concurrent terms separately). A lot of people want to be the President - it's an aspiration. And while you may find it easy to name a few Presidents, it'll probably work out quite hard to name them all...
At first I became a whole city; later on I've been a neighbor, a teddy bear, a shrub and a field of roses. I've been a fat cat that likes lasagna, another time I've been a grant. I've been the son of John, and I've also been a car brand.
This paragraph clues the names of many of the people who have been President in that time:
'a whole city' - George Washington, the first President.
'a neighbor' - OP's intent was Martin van Buren, as 'Buren' is Dutch for 'neighbours' (although personally I also enjoyed @SteveV's suggestion of Woodrow Wilson in comments, paying homage to Wilson the neighbour in Home Improvement!).
'a teddy bear' - Theodore 'Teddy' Roosevelt.
'a shrub' - George Bush.
'a field of roses' - Theodore and Franklin D. Roosevelt (again, a Dutch translation of the name).
'a fat cat that likes lasagna' - James Garfield.
'a grant' - Ulysses S. Grant.
'the son of John' - Andrew and Lyndon B. Johnson.
'a car brand' - Gerald Ford (maybe even Abraham Lincoln too!).
Now I've become the notion that I display my superiority over others, which many others considered true of me anyway since the very beginning.
The current President is Donald Trump - a word whose meaning is similar to having 'superiority over others' (like a trump suit in card games...); a word which his rivals might consider appropriate due to his reputation in business...