Ok Im gonna clarify things. Im pretty good at drawing anime and I can learn new art styles real quick. The problem is I can't come up with an original cartoon art style, in other words I can only copy or draw anime. This may be a problem when I'm applying for university.
tl;dr So If I don't have a cartoon art style will big name studios hire me?
I don't see why not! there's lots of western cartoons that have an "anime" style! Just look at Avatar the last airbender. It was very popular too;)
If you can emulate classic Disney or Warner Bros, then you can be hired anywhere. Those two have long been the defacto "standard".
But understand that when I say "emulate" I don't just mean duplicating an already existing image of a Disney/Warner Bros character--I mean taking those characters and doing all-new poses and situations with them.
And if you want to find out what the big-name studios are doing, a 1/2 hour of searching on Google can tell you who all those studios are in North America, where they are, and what kinds of cartoons they have done.
One clue: you will NOT see a lot of "anime" coming from them.
"We all grow older, we do not have to grow up"--Archie Goodwin ( 1937-1998)
I studied animation in the UK. Mostly for these you had to turn up to an interview before they'd give you an offer/invite you to the course.
There you had to present a portfolio of work to the course leaders, and occasionally all the other applicants as well. As such, I can tell you there is a wide variety of portfolio types which are put forward - and equally a variety are accepted.
Some are looking more for life-drawing based portfolios, others don't mind if you have some animation samples or your own character work. (You just need to demonstrate you can draw, draw well, and know how your work relates to animation.) Most university courses have a drawing module to help you with life drawing, so you can then transfer these skills to recreating your characters in realstic poses and so on.
If you are going in with things that you have drawn in "anime" style, it might be an idea to show them pictures of the original character, as well as how you have drawn them, and have drawn them differently. As Ken Davis said, it is important for them to know you can do more than just copy a picture.
The one thing I do remember our university tutors (who were also actively working in the industry) is that it's all well and good to draw anime, but it is very hard to stand out with it as that style is quite saturated with artists at the minute. (That being said, there were some individuals on the course who opted for this style regardless and did really well.)
So as for will a studio hire you - yes, if you are good and if you can demonstrate you are good. For some projects such as tv series, films etc, everyone on the team will be required to keep to a certain style of drawing so, again, it's a good thing if you can emulate other's art styles and do so well.
Let me also say this as to your own drawing style: looking for it is like looking for your hand at the end of your arm. It's already there - everyone has their own drawing style that is as unique to them as their handwriting. Everyone's is (slightly) different.
Hope this helps a little.
Dear Vicky,
We are looking to hire. Do you have some sample of your work. Please email us at olblue@landofbluebears.com
Thank you,
Birgitta