eSim and Multi-factor Authentication security codes

To get text codes, you'll still need your usual SIM enabled as that's the one associated with your usual phone number. But have data roaming turned off so you don't rack up charges connecting to mobile data.

You'll access personal accounts via the mobile data provided by the eSIM

If you cannot receive a MFA by email, an eSIM will not receive a text from your home cell number. If you have WiFiCalling set up from your provider and can use it internationally from a WiFi connection at your hotel, then you can send and receive texts as normal from your main number. An eSIM will usually only give you a data connection from a local provider as indicated when you connect it to your phone.
You could also use an international eSIM as a data signal to use with your home cell number through WiFiCalling. Both SIM & eSIM need to be active, but turn off data roaming so as not to incur charges from your home provider.

In my experience touring Italy and Spain, most travel eSIMs are data-only and won't give you a new number to receive texts.
To get your security codes, the best way is to use a phone that supports Dual SIM. You keep your home SIM active just for receiving texts (usually free to receive) and use the eSIM for your internet.
Alternatively, switch your bank's MFA to app-based notifications instead of SMS before you fly. This is much more reliable when traveling!

Yarramar said it very well. Here's three more thoughts:

1. If the OP has a cellphone provider or a cellphone which does not support wifi calling, then that option is out. Fortunately, this can be tested at home before leaving.

2. If the OP has a recent iPhone, then the second esim (data only), can be used as a substitute for wifi, so that wifi calling will work. But I have not been able to make this work with my recent Android S24 phone.

3. Another option is to switch the various accounts to doing 2FA via email, if possible. The second esim for data will make email work.