I've found out how to switch between romaji with QWERTY and keitai-style flick keyboard, but I'd like to have ordinary kana layout similar to PC and Mac:
Is it possible with GBoard, or do I need a third-party keyboard app?
Unfortunately, as of the latest version (8.0.5), GBoard only supports 4 variants for Japanese keyboard: 12 keys, QWERTY, Handwriting, and GODAN. The QWERTY variant also only supports romaji input, not kana layout.
The current solution is to use 3rd-party keyboard apps, and while I have browsed some kana/JIS keyboard apps, the only keyboard app that I could recommend for now is:
Japanese Full Keyboard For Tablet by Repy (freemium)
The keyboard supports 3 keymaps: JP109, US, and JP KANA. After selecting JP KANA keymap, the kana mode can be toggled by pressing the 全 key on the top-left.
While the name mentioned "Tablet", it's also working on the phone. However, since the layout tries to follow the physical keyboard layout, the size and the position of some keys may not be optimum for a smaller screen.
Based on a quick test, the keyboard is quite comfortable on a 5" screen while on landscape mode, though on portrait mode the keys are a bit too squished. Fortunately, the settings allow changing the height of the key, so it's still bearable on portrait mode.