2 GFCI or not 2 GFCI
Yes and yes. All receptacles in a bathroom must be on a GFCI protected circuit. You could have the GFCI on the first receptacle in a series string and it can protect others down stream of it, or you can use a GFCI breaker that is dedicated to that bathroom and protect everything. But fair warning, if you tie the light fixture to it (which is not required), then a GFCI trip can leave you in the bathroom in the dark, with the Reset far away!

In your photo, push the Test button on your GFCI and see if the outlet for the bidet seat is also killed by it. I'd be willing to bet it is.