Can we send Monero from an exchange to an exchange directly? Lets say from Bittrex to Poloniex, or do we need an actual wallet and send from an exchange to a wallet first and then from wallet to a new exchange. Is it same with AEON?
It depends. Several exchanges have a requirement for a payment ID to be included when depositing, but have no option to add it to withdrawals.
In general it's safer to withdraw to an own wallet and then deposit to another exchange, since you can track all steps and in case handle out view keys to prove you didn't get or you sent coins.
An exchange will use a wallet just like you use a wallet. The blockchain doesn't care whether you own a wallet or whether an exchange does or whether a robot does or whatever. So, to answer your question, yes, you can send directly from one exchange to another, since they're wallet is like any other wallet.
It would be the same with any cryptocurrency, AEON included.
There's nothing technically preventing it. However, it depends on whether the sender exchange gives an option to specify the payment ID when sending, and implements it correctly. For example, I remember Poloniex had some bad implementation, where even when specifying the PID when withdrawing, it would show in my wallet as having PID=0x0 (maybe it's fixed now, haven't tried in a while). This would not get credited automatically if sent to another exchange. You can test first by sending to your wallet, and see if they're respecting the PID. If yes, you can try and send to another exchange.
The receiver exchange has no way of knowing whether the amount came from your wallet or another exchange (unless they share user data, but that's another story). What's important is that the payment ID is embedded in the TX, otherwise the receiver exchange would have no way of knowing that the payment should be credited to your (or someone else's, if you so desire) account. So really, it all depends on the sender exchange.