No, Penelope is not "Spartan"
.....not in the way you think she is, rather.
People do know that the Sparta from Homeric myths is not the same Sparta as the one we are familiar with in popular culture, right?
People do know that countries, islands and cultures can change over time, right?
The "violent, bloodthirsty Sparta" that people think of (especially the agoge, aka the Spartan education) is mostly documented thanks to later sources, like a source from the 4th century BC, Xenophon (who wasn't even Spartan himself!)
Sparta rose to prominence as a military powerhouse in the 7th century BC. The Spartan education is said to have been created in the 7th or 6th century. Not widely popularised, merely created. So the Sparta depicted in popular culture is most likely inspired by this era of Spartan history.
The Iliad however was written...... in the 8th century BC. Homer was born a century before the military stronghold!
And his stories take place even earlier than his birth. For when does the Trojan War takes place, according to scholars? 12th or 13th century BC!!!!
They're not from the same eras of Greek history!
You're allowed to make headcanons all you want, but it would be severely incorrect to act as if the Spartans from Homeric myths and more generally ancient Greek mythology are the same as their "bloodthirsty, violent and ruthless" counterparts from our usual modern depictions. And that's without even going into the supposed accuracy of this description... (but that's a whole other story)
Six centuries or even more separate Penelope (and Helen) from this epoch.
That's like saying that a British nobleman of the time of the Hundred World War between Britain and France could be accurately depicted as any British man from 2025. Sure, they are both British. From the same city even, perhaps! But they are not... exactly... culturally similar...