Wordorigins.org

No, the phrase spick and span is not related to the ethnic epithet. It is an adjective meaning perfectly or brand new or a reference to refurbishing or cleaning that restores something to mint condition.

The original form of the adjective was span-new, a form that survived in dialectal speech into the 19th century. The word is from the Old Norse spán-nýr, literally meaning chip new, as in new as a chip just chiseled from a block of wood. From The Lay of Havelok the Dane, c.1300:

Þe cok bigan of him to rewe, and bouthe him cloþes, al spannewe.
(The cook began to pity him, and bought him clothes, all span-new.)