If the phone originally came (new) from Koodo then they might have preconfigured it to always display "Koodo" instead of allowing it to display the ICCID.CellInfo.CellIdentity.SystemService text encoded on the SIM card. (Apparently there's some kind of "Issuer Identifier Number" registry and database which tracks these things but I know nothing about it.)
I've seen "Public Mobile", "Public Mobile 3G", and "Public Mobile LTE" displayed before. I've also often seen "TELUS" displayed after phone restarts, usually for days at a time. It really doesn't matter what's displayed as long as it works.
Alternately, it might be displaying the APN Name. You can use whatever text you like when you create a new APN (or edit an existing APN) with Public Mobile's APN settings. And you need to do this to access data (and MMS attachments) through Public Mobile anyhow.
Some devices (particularly Android 8 devices from a few years ago) do not allow users to add/edit APNs - the option is locked and "greyed out". There are a variety of simple workarounds and complex technical fixes for this old problem but none which always work on all devices.
Apparently there used to be problems on some devices with Public Mobile's default display name being too long. And apparently this has been corrected (by Public Mobile or by newer devices).
Transitions between Telus, Koodo, and Public Mobiles are usually unproblematic. Or so most people around here seem to say.
https://www.youtube.com/watch?v=cknxcmZnln4
https://forum.xda-developers.com/t/carrier-name-in-today-screen.293614/https://stackoverflow.com/questions/34714656/how-to-get-carrier-name-from-dual-sim-phone-android
