Does every Galaxy have a black hole? Is that what keeps galaxies a Galaxy?

 There are many Galaxies with no detected black holes (BH). However, as the saying goes, absence of evidence is not evidence of absence. The fact is that it’s very hard to detect any single object at galactic distances unless it’s extremely luminous. In fact, we can only detect those BH’s that produce powerful astronomical jets. These jets are made up of a collimated beam of high-speed particles (see illustration below). To produce these jets a BH must be actively “swallowing up” large amounts of matter. But in older galaxies the BHs already swallowed everything in its vicinity (i.e. cleared its neighborhood). So there is no jet and they become nearly undetectable.

However, this is not a permanent state. If we wait long enough (perhaps thousands or millions of years), an unlucky star or nebula will eventually stray too close to the BH, fall into it, and produce a new jet.

Q. Is that what keeps galaxies a Galaxy?

The OP’s second question is much easier to answer: A typical galaxy has many millions of BHs of various sizes, among them is often a supermassive BH at the galactic center. As far as we know, we can remove all of them, and almost nothing will change in the structure or dynamics of the galaxy. They are just there, adding some small fraction of gravitational pull, but not much else. They are not needed to keep the galaxy a “Galaxy”.