Yes, push notifications are more secure than SMS.
An SMS can be sent to users without their consent. There is a history of data breaches using SMS where the users can land in a potential malicious trap of hackers. In addition to that, it is very easy for hackers to send messages to your mobile devices.
On the other hand, push notifications are authenticated much stronger with user consent as the priority. Other benefits are multi-factor authentication for mobile apps and Google takes it very seriously to block sites indulging in spam notifications through regular browser updates, and more.
Sign-up to Truepush, a free push notifications service with a monetization plan that streams additional revenue.