Yes, typically they are. However, if you go look at the definition and derivation of the word “soldier” you will find it has several meanings, in fact. The derivation is from France circa 1300...essentially “service for pay”.
I used the word in the context of serving ones country in an armed capacity; that applies to other forms of service and specialties that directly or indirectly support armed operations. In my book, they are all ‘soldiers’.
Lastly, I think that there are several special units from both the Navy and the Air Force that would bristle at the thought of not being a ‘soldier’. The SEALs are one, Air Force Special Operations is another (you know the guys that ride in on camels and donkeys to light up a target for the GBUs) - there are many other such units.
The few SEALS I know are certain they are sailors.
Roman marines got paid half the wages of the Roman soldier.
A soldier is one who fights as part of an organized land-based armed force;
per wikipedia.
Now some would argue the AF doesn’t qualify because of organization, not its land base, but I wouldn’t go that far.