How do you handle bank inquiries while abroad if using eSim vs international Consumer Cellular?

Sounds like using an internatiinal eSIM you will have to add your international eSIM number to your bank account and indicate that is the number to text... if they will even text to international numbers.

How many accounts other than your bank might need to message you for access or to retrieve a password or ???

Will/can this be avoided if adding the international plan to my
existing Consumer Cellular plan?

Yes

Thanks for the reply. I will go with the international CC plan. I doubt that I will be spending much but who knows. I started to question this after my ETA application for the UK was stopped until I responded to a delayed text asking for verification.

There is also a category of eSIm's that are data-only. You would use those in addition
to your existing voice/text service, which might negate the cost benefits of an eSim in
the first place, but in this case, you would then have the same telephone number as
when in the US.

My bank sends alerts by text snd email. Maybe see if they can email you.

The other issue that can come up is US websites that are not accessible with other than US IP addresses (European eSIM). Doesn't happen terribly often, but it tends to be the case for things like government websites like my Texas drivers license replacement site and my professional registration site ..... and some shopping sites like Eddie Bauer. When I had cable TV in the US, I could access the channels online with my US IP phone, but not my Hungarian phone. This will probably have no impact in you ..... unless it does.

I also like that with my regular phone number everyone knows how to reach me in an emergency.

Shanah Tovah U'Metukah

I use eSIMs overseas. Because they cover data, any apps on my phone that provide notifications, work just fine. If your credit card company or bank has an app, enable notifications and you should have no difficulties.

I simply get all my notifications by email. Chase, Amex, Schwab. Same as I do at home. Shows up almost instantly. A SIM card allows email, so there is no problem.
I do not know what you mean by bank asking for verification of a transaction. I have never had that happen, at home or overseas. I have never had a card denied.

Some banks under certain circumstances, either bank initiated or customer requested, will request a code to complete the transaction. My last RyanAir purchase had this. A window popped up and asked for the number that the bank had just texted me. The pop-up window pulled the number out of the text and completed the transaction without me having to do anything. Sometimes, I have to manually enter it. I dont mind. It isn't often, and its generally larger transactions and its another layer of protection. Im okay with it.

treemoss2-when I was booking my flight and UK hotels, I needed to provide a code (sent to me via text) acknowledging the transaction as valid. I was also sent an email alert. Ditto with the ETA application.

Since I rarely look at emails during the day, I would have missed the acknowledgement. A text I will see earlier but I am NOT someone who hangs by their phone (usually it is sitting on the counter) and seldom look at emails on my phone. Hence my inquiry.

I have gone with the international plan, keeping in airplane mode with data off until/unless needed. May be overkill but...

I have never had that happen. Ever. And since you are not always monitoring your phone (me neither) you may miss a code and then lose whatever reservation you may have made. I would contact my CC company and question this.
Even if you have an international plan, the problem remains.