No it doesn't. The birthday paradox occurs because the size of the sample set is large relative to the universe.
For example 15 students is a lot relative to 30 potential birthdays.
However even if we had 1 billion users with 1 billion wallets EACH and each wallet used 1 billion private keys it would be less than 1 in 1/1,461,501,637,330,900,000,000th. Of course we don't have a billion users and they don't have a billion wallets each and the average wallet hasn't used 1 billion keys so the odds of a collision are quadrillions of times less likely.
Birthday paradox only applies in a meaningfulway if the size of sample set is large enough to be within a couple magntidudes of the universe.