I was wondering if the SameSite flag on the session cookie was enough of a protection against CSRF attacks.
I see CSRF token solution everywhere, but I am not sure about the need to use a CSRF token if the cookie used for authentication is already protected by the SameSite flag (in Strict mode).
On top of that, if I understood it well, tho cookie would still be sent along with subdomain URLs like api.myapp.com which would be perfect for my needs.
This is somewhat opinionated, or at least depends on your target audience and risk appetite.
SameSite=strict is supported in almost all fairly recent browsers as seen here, but note the exception of IE11. Not many people use IE11 anymore, but for them it will not be good enough. Only you can answer whether that's good enough for your usecase, as of writing this, a significant amount of users would not be protected.
Also the general consensus seems to be that SameSite should only be used as defense in depth (eg. here or here in a similar question), but most of the concern is around Lax, and less about Strict. However, Strict is very user-unfriendly, in a real-world application you probably can't really use Strict, because that's very bad UX.
The usual arguments are around browser support (as above), GET requests changing state (only relevant to Lax), and some special cases still revolving around GET changing state.
So my take currently is that the reason SameSite=Strict is not good enough in general is the lack of full browser support (IE11), and a strong point against it is bad user experience. I can imagine circumstances where it is good enough. SameSite=Lax I think is only a defense in depth measure, because of the issues above, which probably don't affect your application now, but might in the future, and nobody will remember to think about SameSite settings.