See the correction below.
The class of games does indeed constitute a commutative ring under Conway's definitions of sums and products. This is already in Conway's book, but more explicitly stated and proved in Combinatorial Game Theory by Aaron N. Siegel, American Mathematical Society, 2013. On the other hand, the theorem that ensures that the cancellation law holds for multiplication is explicitly formulated and proved for surreal numbers (Theorem 8, pp.19-20 for Conway; Proposition 2.3, p.413 for Siegel). I suspect it fails for games.
Correction (1/13/22): (Siegel p. 413) establishes the following for the multiplication of long games. For all long games $x,y,z$: (a) $x0 \cong 0$, (b) $xy \cong yx$, (c) $x1 \cong x$, (d) $(xy)z=x(yz)$, (e) $(x+y)z=xz +yz$. Siegel's long games are Conway's games.
However, as David E Speyer observes in his comments, contrary to what I previously claimed this is not enough to show that the class of games constitutes a commutative ring under Conway's definitions of sums and products. See Speyer's comments for details.