iPhone Dual Sim Incoming Call identification trouble

Re: "... I have no idea whether it’s a business call or a personal call, because there is visible one label (of the eSIM) ..."

Answer:

The incoming lines are identified by the small "box" indicator with a single letter in it - displayed immediately ahead of the CallerID. - .

The letter displayed in the box is simply the first letter of the "name" you've assigned to the line in Settings-> Mobile[Cellular]

(e.g. P=Primary, A=Alternate, A=AT&T, V=Verizon, W=Work, P=Personal, S=Secondary, ... etc.)

This "identifier" is displayed on the incoming call "pop up" and is also displayed in the call logs.

Commentary:

IMO, this is one area presenting Apple w/ an "opportunity for product improvement."

Visual Indication -- The incoming lines could be have more distinctive indication (e.g. Color vice the small B&W box w/ a single letter letter)

Audible Indication -- Clearly the ability to set a different ringtone for each line would ba a HUGE benefit.

Desired improvements aside ... once you become accustomed to looking for the "Indicator Box" in addition to the incoming CallerID ... it's certainly "functional" even if not [yet] everything we might like to have.

See also: Using Dual SIM with an eSIM - Apple Support

Suggest you might provide product feedback re: this to: Feedback - iPhone - Apple

Look for "Recent" indication as shown in the pic below either against a Phone number or Apple ID. This means either Apple ID was used or a Phone number was used to send the iMessages.

For SMS/MMS it will always be a Phone number only.

Have you set different names for each line in “settings” ?

e.g. in this example the owner might have his two lines labelled “Primary” and “Secondary”