How many wallets a person can download and use?
As many as they want.
If I used a wallet in the above example, then I wanted to change my wallet, should I download a new one and do a transfer from my previous wallet to new one?
Yes that is often the best method, but that will incur a transaction fee.
I mean there is no unique ID for people that they can use it in different wallets to reach one unique account?
Yes there is a unique piece of data that can be moved between different wallets and which gives users of that wallet control over an amount of Bitcoin. This is the private-key. It can often be generated from a seed-phrase, sometimes called a recovery phrase.
You could have ten wallets each containing the same private key. All of those ten wallets would show you the same "balance" and any could be used to spend all or part of that "balance". The other nine wallets would find out about the transaction and adjust the "balance" they show. However having your private key in ten places is less secure than only having it in one (plus an offline backup of some sort).
You can alternatively import private keys into a new wallet but you need to understand the difference between the early type of private key and those used in hierarchical deterministic (HD) wallets - an "extended" private key from which others are generated. When moving a private key from an HD wallet into another you need to make sure the new wallet is configured to use the same "derivation paths". This can be complicated so it requires care.
How crypto exchange websites work?
In my view, exchanges are mainly for exchanging one currency into another. For example USD into Bitcoin or Bitcoin into Litecoin etc.
A member of an exchange will deposit some money into their exchange account and can then typically transfer some to the exchange account of another user of the same exchange without any fee. Amounts transferred this way are not recorded in the Blockchain. The exchange remains the "owner" of the money.
Exchanges also typically allow members to transfer money to non-members. These are typically a transfer from the exchange's wallet to the other persons wallet and this transaction is recorded in the Bitcoin blockchain and involves a transaction fee.
How my personal wallet will contact the exchange's internal wallet?
Wallets broadcast intended transactions to a few nearby nodes. They pass it on. Eventually a miner incorporates that transaction into a new block and broadcasts that block to nearby nodes. They pass it on. Eventually your wallet and the recipient's wallet hear about the new block. Eventually the wallets mark the transaction as confirmed and update the calculated "balance" they show their users.
It is worth remembering that Bitcoin are not stored in wallets or in the Blockchain. Wallets store private-keys. The Blockchain stores historical records of transactions. A transaction is a record of a transfer of control over some amount of money where Bitcoin is the unit of measure.
When you give money to an exchange, you no longer have money, you have an IOU.