Some magic items, such as the Circlet of Blasting or the Eyes of Charming, allow you to cast a spell, but specify the attack bonus or save DC. Other magic items, such as the Robe of the Archmagi and the Artificer's Enhanced Arcane Focus increase your spell attack bonus and/or save DC. Do the latter kind of items increase the fixed values of the former kind?
They do not interact
As you’ve already stated in your question, items such as Robe of the Archmagi have the effect of increasing your spell save DC:
(…) Your spell save DC and spell Attack bonus each increase by 2.
When casting a spell through an item like Eyes of the Charming you are not using your spell save DC, you are using a fixed value that’s provided by the item and can neither be increased nor decreased:
(…)While wearing them, you can expend 1 charge as an Action to cast the Charm Person spell (save DC 13) (…)
Your spell save DC is a term with mechanical meaning in 5E, generally only spellcasting characters have it and the equation for it is given in your Spellcasting feature.