When you catch the first 10 fish in Fishing Clash, a window with the possibility to add your custom name appears. The option to change your name is not available before that.
You may change it later through your profile, by tapping on the profile icon in the top left corner of the main screen.
Please note that names with profanities, foul, or abusive language are prohibited. Entering such names will lead to your name being changed by the Fishing Clash team.