No. In the world of Narnia magic can be learned (and used in a haphazard fashion) by those without magical blood. In the world of Harry Potter, it's exclusively used by humans with magical ancestry.
"I see," she said scornfully, "you are a Magician—of a sort. Stand up,
dog, and don't sprawl there as if you were speaking to your equals.
How do you come to know Magic? You are not of royal blood, I'll
"Well—ah—not perhaps in the strict sense," stammered Uncle Andrew.
"Not exactly royal, Ma'am. The Ketterleys are, however, a very old
family. An old Dorsetshire family, Ma'am."
"Peace," said the Witch. "I see what you are. You are a little,
peddling Magician who works by rules and books. There is no real Magic
in your blood and heart. Your kind was made an end of in my world a
thousand years ago. But here I shall allow you to be my servant."
The Magician's Nephew - CHAPTER VI
That in itself indicates that these are not set in the same fictional universe.
With regard to the similarity of the lions, you should be aware that both the Narnia shields and the lion depicted on the Gryffindor banners are not original creations. They're representations of a heraldic "Lion Rampant" which goes back to at least the 1400s (and before). Most depictions after the 1800s follow an extremely similar pattern to those seen in various fantasy books/films including Narnia and Harry Potter
Arms of Strange and Talbot - Catalogue of the Heraldic Exhibition in London, 1894.
You might wish to note that JKR spoke on the key differences between the world of Harry Potter and Narnia
I found myself thinking about the wardrobe route to Narnia [in the C.S.
Lewis series including The Lion, The Witch And The Wardrobe] when
Harry is told he has to hurl himself at a barrier in Kings Cross
Station - it dissolves and he's on platform Nine and Three-Quarters,
and there's the train for Hogwarts. Narnia is literally a different
world, whereas in the Harry books you go into a world within a world
that you can see if you happen to belong. A lot of the humour comes
from collisions between the magic and the everyday worlds. Generally
there isn't much humour in the Narnia books, although I adored them
when I was a child. I got so caught up I didn't think C.S. Lewis was
especially preachy. Reading them now I find that his subliminal
message isn't very subliminal at all.
Really, C.S. Lewis had very different objectives to mine. When I write, I
don't intend to make a point or teach philosophy of life. A problem
you run into with a series is how the characters grow up ... whether
they're allowed to grow up. The characters in Enid Blyton's Famous
Five books act in a prepubescent way right through the series. In the
Narnia books the children are never allowed to grow up, even though
they are growing older.
"The story behind the Potter legend:
JK Rowling talks about how she created the Harry Potter books and the magic of Harry Potter's world,"