Does copying the book's figure for my dissertation fit the copyright?

In fact, I think that your proposed use would not fall under "fair use", in the sense that you are not addressing the other authors' work, but merely (!) using some of their stuff. The fruits of their labors... which is what various reasonable notions of intellectual property are meant to protect.

Of course, if you write the copyright owners and they say "go ahead", you are legally fine...

However, legal correctness and moral/ethical correctness are not the same.

Might be better to take the trouble to make your own graphic that is good enough. At the very least, in addition to investigating literal copyright issues, you might contact the people who made the graphic and ask their permission. If everyone says "go ahead", out of generosity or whatever, then you are truly fine.

If you are critiquing the image or expanding on what was said then technically you are in the clear. In the real world it comes down to what Springer decides and the country that you are in. The exemption there is not really an exemption in the US. That is copyright law, the part called fair use. How the courts see it if you were sued is a different issue and Springer has the deep pockets here and could make your suffer no matter what. Whether it is worth it to Springer depends on the circulation of your thesis. If you can reproduce the graphic yourself you will obviously be better off.