Will the high volume of transactions make Bitcoin more anonymous?
No. For example, let's assume, that you make a transaction like this one:
https://mempool.space/tx/000000000fdf0c619cd8e0d512c7e2c0da5a5808e60f12f1e0d01522d2986a51Then, does it matter, how many transactions are there? In practice, not really, because for example making a transaction with a lot of leading zero bits is something, where you have to write some code, to achieve that. Which means, that a typical user won't do that, and it is crystal clear, that someone who did it, had some coding skills. And that information alone can be used to narrow down the possible set of coin owners.
I mean those transactions that use mixers, CoinJoin and btc to monero exchange methods to hide the identity.
If you have a lot of small transactions, and some huge CoinJoins, then CoinJoin participants are only hidden inside the CoinJoin traffic. That's why, it is possible to sometimes achieve better results, just by doing a regular one-input-two-outputs transaction, than being a part of some huge CoinJoin.
The same with address types. Some people use only addresses starting with '1'. If you increase Segwit traffic, it won't change their situation (and vice versa).
And the same with coin amounts. If you send 0.01 BTC, then you can hide in a group of people, which also use round amounts. And seeing more transactions, sending 0.01234567 BTC, won't change your situation.
I mean hundreds of millions of transactions every day
If you want to get there, then you need transaction joining. And then, things which can be seen on-chain, are loosely connected with actually performed transactions, because then, a single on-chain transaction can handle a lot of users. Which means, that you have "Alice -> Bob -> Charlie -> ... -> Zack" seen in wallets, but "Alice -> Zack" is what you can actually see on-chain. The 4 MB witness limit won't let you push hundreds of millions distinct transactions on-chain, where every user would have its own UTXO. Instead, you would have multiple users per UTXO, to fit all of that in a block.
with millions of transactions, it should be very difficult to track, right?
Difficult, but still possible, because the traffic of unconfirmed "to-be-batched" transactions, can still be observed. For example: if you have Lightning Network, then on-chain, you have just some channel opening, and channel closing transaction. But: if you run some LN node, then you can observe, how coins are flying through your node, and you can get some statistics, out of that. And then, on-chain, after getting a lot of confirmations, you can only see a few transactions here and there, to set up, and close channels, but if you had a node, running 24/7, then you can have a clue, what happened in-between. And the same is true with transaction joining: nodes would still need to broadcast to-be-batched transactions in unconfirmed state, and they could be traced in a similar way, as LN transactions are traced today.