A cursory US trademark search for contactless payment marks predating Apple Pay, finds examples like Pro Pay, Easy Pay, Smooth Pay, Monitise Pay, O Pay, Isis Ready Pay, Express Pay, Tap & Pay, Mastercard Pay. And of course "Loop Pay" (now owned by Samsung) was also publicly used before "Apple Pay".
Not to mention that Google came out first with Google Wallet, Samsung came out with Samsung Wallet... then Apple started using the name Apple Wallet.
So should Apple have avoided using "Pay" and "Wallet" because others had already used those words? Of course not. Common terms help users and brands alike.
In short, Apple didn't invent using "pay" for electronic payments, or "wallet" for a virtual wallet, any more than Microsoft invented "windows" for a windowing GUI. Now, if Apple had called their method something non-obvious like "Apple Air Dollars" and someone else afterwards used "Air Dollars", then people could give them grief. Otherwise, no.
When Apple Pay came out, it was noticed that it had a similar look, listed the same merchants, similar APIs, similar partnerships, and even the same method of entering a card via camera, that Google Wallet had.
Even the first Apple Pay web page looked very much like the Google Wallet webpage.
This is to be expected with similar purposes.