After I broke up with my girlfriend about two months ago, I am still in contact with her, we talk on the phone sometimes, talk when we see each other in uni, online and lately we sort of go out not quite as a date though..

Yesterday we went out, we talked, we laughed it was a good time. We also kissed too, she hugged me and kissed me back, I told her I still love her and I know she loves me too and wants to get back together again.
We talked about marriage and having kids again, and all that time she was smiling and how I know her, I think she still wants that with me.

But yesterday after we kissed and hugged, hold hands, she didn't say anything about getting back together after what happened that day.
My question is should I ask her are we back together or should I wait for her to tell me on her own ?

First, we need to know how old you two are -- 15, 25, 35? >remind me to check the OP's other posts<

Have either of you dated others now or before?

You are in university right? So that puts you at at least 18.

Take it slow. It's only been 2 months since the break up.

We're both 20, and we both dated once with other people before dating each other.
After we broke up we haven't dated nobody else..

Why did you break up?

I was jealous, didn't want her to hang out with guy friends and when sometimes she lied to me I confronted her and she would tell me, but if I hadn't known about it she wouldn't tell me.
So one day she asked for a month break, but that turned out into a breakup after a mutual friend of ours told her lies about me which made her decide to break up..

But after she found out that she (the friend) was lying to her she started talking to me again, and apologized that she didn't believe me and believed her lies..

You need to take this slow and both of you need to work on your relationship skills. Jealousy is a total and complete deal breaker. So is lying.

The two of you need to learn to communicate as communication is the key to any relationship.

It's too soon after the break up to ask her to be exclusive and visa versa. The two of you need to talk, get to know each other.

You need to stop the jealousy and she needs to stop the lying.

You need a clean break (no contacting each other or hanging out) before you should even think about getting back together.

From what you've described, your relationship ended in name only. The amount of contact may have lessened, but it was still there. Neither of you have had a chance to register, beyond the most basic and superficial level, that you're no longer a couple. Until that happens, you're not going to be emotionally able to judge if you SHOULD be together.

So we got back together since yesterday officially but there's just one small thing...
She said that she loves me and wants to marry me and other stuff but I feel like its not the same like it was.

She used to care a lot more, we talk on the phone more often but yesterday I asked her will all things be the same as before, she said no, but I don't know if she meant it in a good way or a bad way.
She doesn't say I love you or I miss you only when I ask her that..

Maybe I'm a little paranoid, maybe it's it's still too early and she can't say that those things immediately but I'm only asking for some advice.
This is the first time I really felt this much for someone and I'm afraid of breaking up again with her..
What do you guys think I should do next ? :/

So we got back together since yesterday officially but there's just one small thing...
She said that she loves me and wants to marry me and other stuff but I feel like its not the same like it was.

She used to care a lot more, we talk on the phone more often but yesterday I asked her will all things be the same as before, she said no, but I don't know if she meant it in a good way or a bad way.
She doesn't say I love you or I miss you only when I ask her that..

Maybe I'm a little paranoid, maybe it's it's still too early and she can't say that those things immediately but I'm only asking for some advice.
This is the first time I really felt this much for someone and I'm afraid of breaking up again with her..
What do you guys think I should do next ? :/

You should NOT smother her. If you are constantly asking for an "I love you" or asking "Are you still mad at me" you will smother her and drive her away. Just take it slow. You can't control how she feels, what emotions she is having, or her actions. Just be yourself, let things develop, and if its meant to be it is meant to be. Things may not be the same. Things may even be better than before. But not if you appear to be a needy boy. Be a man and let things happen.