While it is possible that there were jokes made ( Tudor England is not my specialty) a larger point would be the amazing lack of imagination for first names in pre-industrial England. As you can see in this handy table here the most popular name was Elizabeth, followed by Margaret and then Jane. Anne is 6th on the list.

Under every name you will also notice variations, beyond spelling ( which was itself quite varied). Not only would it not have struck anyone that it was odd for Henry to have married a couple of Janes and Annes, that lack of imagination meant that it was very common for just one household to have a few generations with the same first name. To avoid confusion, people would use diminutives or abbreviate ( so, William, Will and Willin: Margaret, Meg, Maggie) or indicate the generation ( so, Nicholas, Nicholson). Under one roof there could be Old Richard, Dick, and Dickson. Obviously, some of these names ( like Wilson and Nixon) eventually acquired a status of their own.

Bardsley, Charles W. (188) Curiosities of Puritan Nomenclature. Chatto and Windus. https://www.gutenberg.org/files/39284/39284-h/39284-h.htm