Yes, like others have mentioned the FP icon does move ever slightly as AOD refreshes.
I cannot confirm this 100% but when my phone is in "unlocked" mode meaning whenever I use boxby routine to have my phone unlocked at home, the FP icon seems to disappear. Let me know if anyone can confirm this.
Not sure but if you have Samsung Pass activated with FP then you will probably not have the option to turn the icon off from AOD unless you put the AOD itself on "Tap to show" or "Off" mode.
As it has been explained by the German member, you need to access to the fingerprint settings (setting page used to enter new fingerprints, which requires to type in your pincode).
On that page, below the fingerprints already registered, you'll see an option to display the fingerprint icon, something like "display fingerprint icon on standby" -> choose "never" -> issue solved