You might have already seen this, but please take a look at the "How to change the main company or site caller ID name" section in this Zoom Support article. Note that it can take a few weeks for the Caller ID info to propagate through the various telephone/cellular companies.
It's been a few months since I moved over to Zoom Phone, but I know my Caller ID does come through with most carriers now. Seems like some of them just don't bother to show the Caller ID at times.
If you try working through that support article and still don't see any success, try sending a Support Ticket to the Zoom Phone support staff from this link, and select Zoom Phone from the Product dropdown.