I must say, it is indeed a little close to the Spanish araña and the French araignée, (and of course, the Latin aranea or the Esperanto araneo, if you wish). Seems superficial, though — I can't find a connection. The Turkish word apparently comes from Old Turkic örümçek meaning spider but literally meaning weaver, and the Latin aranea is possibly from the greek arákhnē. We know of the legend of Arakhne the weaver who was turned into a spider by Athena. Is arákhnē related to örümçek? There is less resemblance between those than between örümcek and aranea (when pronunced). So perhaps not.
A good way to test it is to see if the Turkic languages spoken further away from Europe - such as Kazakh or Uyghur - have a similar word for spider. If they don't, it's more likely that a Turkish word has been influenced by Turkey's neighbors. In this case though, the resemblance is fairly superficial, so it's likely a neat coincidence :)
Thank you guys for checking. If it ever existed, my guess is that it was an obscure Ottomanism. As you know, it was fashionable to use Perso-Arabic vocabulary and constructions in the most "sophisticated" literary and poetic levels of the language during the Ottoman period. I studied Arabic and Persian, so I'm always on the lookout for borrowed elements in Turkish ;) Teşekkürler!