Are you referring to Azure Information Protection labels or M365 sensitivity labels? The question is tagged with AIP but you say Microsoft Information Protection sensitivity labels in the question. Note that AIP labels and sensitivity labels are defined in different places: https://techcommunity.microsoft.com/t5/office-365/what-is-the-difference-between-sensitivity-labels-and-aip-labels/m-p/332109
When you create the first sublabel for a label for an AIP label, users can no longer select the original, parent label. https://learn.microsoft.com/en-us/azure/information-protection/configure-policy-new-label
There is a troubleshooting guide that goes over potential issues with AIP labeling. Usually an issue would be related to user permissions. https://learn.microsoft.com/en-us/troubleshoot/azure/general/troubleshoot-aip-issues
This article goes over potential issues when editing sensitivity labels. Until you publish your labels, they won't be available to select in apps or for services. To publish the labels, they must be added to a label policy, and it may take 24 hours for the new labeling to apply.
Also, for the Azure Information Protection unified labeling client the label ID remains the same even if you change the label name later on. https://learn.microsoft.com/en-us/azure/information-protection/rms-client/clientv2-admin-guide-customizations
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