The tiles are marble and the dark inlays are granite
Its likely that when installed the marble had a nice polish but traffic and harsh chemistry has dulled it.
To "just" clean it stick with non peroxide based , pH 7-9 carpet or Hard Surface pre sprays and rinse with water.
If you go higher or lower you'll likely dull it even more.
The grout is pretty dirty so use a well chiseled pole brush on the lines, a
CRB or 175 typically wont get into the recession well enough.
Turbo spinner at 700 or HS wand at 500 should do the trick
White pad or MF buff when dry for an even appearance
There's probably a lot of etching (dull areas) from spilled acidic drinks and chemistry present, so make sure to let them know(in writing) that cleaning alone does not correct that.
The granite area are not a concern.
You might want to learn how to use DIPs (diamond impregnated pads) to put a shine or gloss (and remove most etch marks) on marble, travertine and limestone floors if you want more of this type work.