do I need to have a second phone number for an eSIM

Yes. You cannot use your existing number under your existing service with another carrier unless you port that number to the other carrier. That would close your service account with the original carrier and move your number and service to the new carrier.

So to use an eSIM with a different carrier and keep your current service as well, you must get a second telephone number for the new carrier and service.

If you just want to change service, then you can contact the new carrier and ask them to help you port your current number from your current service. The new carrier can set that up - your number would remain unchanged but your old service will be cancelled and your number would move to the new service with the different carrier you want to use. But note you want to ask the new carrier you wish to use to initiate the number port from your old service.