I cannot find a proper definition of multivariate martingale. If each component is $1$-dimensional martingale is it enough for a $d$-dimensional process to be a martingale?
You mean the parameter set is still one-dimensional, and the values are in a vector space like $\mathbb R^n$? Then yes, the definition can be that each component is individually a martingale. Or abstractly that when we apply any linear functional, the result is a scalar-valued martingale.
"Multivariate martingale" might also mean a situation where "time" is replaced by a multi-dimensional parameter set.