They reuse names for several reasons; it’s their given name, honor an ancestor or glom onto the legacy and cache of the name.
If Ruler II was popular and considered good, there could be incentive to become Ruler III and leverage the positives associated with it.
As for picking a new name, they could, but most tend to be based on characters from their respective religions and belief systems. For the UK, the ruler is also head of the Church of England, so it wouldn’t be popular for them to assume Ahmed I as a ruling title as it’d alienate them from their subjects.
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