In the same way/manner/fashion all establish a similarity of how something is done, said, thought, etc. The more general in the same way can be used in any register; manner and fashion are slightly more formal and more likely to occur in written rather than spoken English. For the most part, all three are functional equivalents, though in the same fashion is less likely to be met in constructions such as _in the same way/manner of x unless one is speaking of literal fashion, i.e., clothing.
In the same spirit usually implies a similarity of idea, thought, design, motivation, or emotion:
She read Dickens in the same spirit she would have eloped with him. — Eudora Welty
With the object of lightening the burden assumed by countries granting asylum in the first instance, other States shall examine in the same spirit of solidarity the appropriate measures to be taken...
In 1944, the International Labour Organization laid out its “Declaration of Philadelphia,” a full-fledged social bill of rights in the same spirit as FDR's State of the Union address of the same year.
Eudora Welty might has said in the same way, but what she means is that the woman in question approaches Dickens in the same way she might have entertained a whirlwind romance, a notion which way might not have easily conveyed. In none of the other examples could one substitute way, manner, or fashion.