UI Builder element name not updated when copy-pasted after a save has been made to the UI Document

How to reproduce:
1. Open a Unity project
2. Open or Create a UI Document
3. Add any element to the hierarchy
4. Save the UXML
5. Rename the added element from the context menu
6. Copy and paste the element

Expected result: Element is copied with the new name
Actual result: Element is copied with the previous name

Reproducible with: 2023.2.0a9, 2023.2.0a16
Not reproducible with: 2020.3.48f1, 2021.3.26f1, 2022.2.21f1, 2023.1.0b18, 2023.2.0a8

Reproduced on: Windows 10

Notes:

• Saving the UXML makes the copied element have the new name
• Does not reproduce when renaming from the Inspector
• Forum post: https://forum.unity.com/threads/ui-builder-incorrect-element-name-when-copy-pasting.1439113/#post-9027559