Do blasé and unfazed have a similar meaning?

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Quite so.
"Blasé" means being used to something to the point of apathy or indifference because one's seen it too much.
"Unfazed" means unaffected, untouched, not registering any impression from something.

However, there's a slight yet essential difference. (blasé is a hyponym of unfazed; it's more specific)
If you're blasé about something - you're unfazed by it due to the amount of times you've seen it already.
But if you're unfazed by something, it might be because you're blasé about it, or it might just be because you don't care about it or because it doesn't concern you.