Poul Anderson's Manson Everard meets King Hiram of Tyre whereas SM Stirling's Ian Arnstein meets Odysseus: Biblical and Classical references, respectively. There is another difference. Everard meets the Hiram of the timeline that is guarded by the Time Patrol. This is not identical with our timeline because to us Sherlock Holmes is a fictional character. However, the two Hirams should be indistinguishable.
By contrast, Arnstein meets Odysseus in the year 10 AE of a very different timeline. But he is still Odysseus. We are used to encountering different versions of fictional, historical and historical fictional characters in novels, films and TV series. Now we need to add "timelines" to the list after novels etc. Indeed, one Time Patrol installment shows us two actors playing the role of the Biblical/historical Cyrus the Great in otherwise indistinguishable timelines.