The Germans had a handy reference chartthat spelled out who was and was not a Jew based on the ‘Nuremburg Laws.’ The chart goes back three generations. This shows the first big problem with trying to falsify your ancestry. For such a deception to succeed, the escapee had to provide documentation all the way back to their great-grandparents. And if you conveniently ‘lost’ your documents, the municipal government would surely have kept copies, and there were plenty of former ‘friends’ in the village who already knew who you were.
The second big problem was that the Holocaust was not a sudden event. Nobody just showed up one day and said, “Hey, we’re going to kill all the Jews now, so show us your papers.” The Holocaust happened incrementally and it escalated over the course of the decade. The Nazis started researching and databasing these family records years in advance. Most people’s “status” was probably decided long before they ever realized it was a matter of life or death.
So what if someone did just run away and move to a new city and try to erase their past? People were not as mobile as they are today. If you showed up in a new place before the Nazis moved in, there were no shortage of collaborators who would happily turn you in. To be fair, there were many people who did try to resist and many who did help the Jews. But when a Nazi investigator turned up, it didn’t take much effort to identify that one suspicious foreign family who moved into town and doesn’t like talking about their past. And if they didn’t have their “identification papers” (which were crucial documents) it wasn’t hard to put two and two together.
So does that mean that nobody ever succeeded? No. But it wasn’t easy.
Solomon Perel is a pretty well-documented example of a Jew who successfully ‘passed.’ He somehow convinced his interrogators that his identity papers had been destroyed. He entered the German Army. When a German medical officer tried to rape him, the rapist discovered Solomon’s circumcision but decided to (A) not rape him and (B) help him rather than turn him in. Weird. Anyway, he successfully maintained the deception. I think they made a movie about him.
There are many other examples of people who hid themselves or established new identities. You can Google it and find all kinds of anecdotes about people who came up with various schemes to avoid the Nazis. Practically all of them needed the help of an ally who could vouch for them. Often, it involved parents giving up their children to friends who would adopt them and help keep their secret. (The parents, of course, had no such option and knew they would die). This was extremely risky for all involved.
But to return to OP’s question: How many attempted to hide their identities and how many succeeded? Nobody knows. There are many stories of people who lived their whole lives as Catholics and only confessed on their deathbeds (long after the war) that they were born “Jewish.” There are organizations (such Shevai Israel) dedicated to helping descendants who want to re-claim their Jewish identities. I read one article that claimed experts estimate ‘tens of thousands’ of people either lived undercover or were raised without knowledge of their Jewish heritage. But that’s the problem with keeping secret… It makes the phenomenon really hard to quantify.