On a comment chain in /r/Europe, a comment asserted that the 3 million Polish Jews were not integrated into Polish society prior to the Second World War. The comment cites a source written in 1944, which I'll quote below:
You can ask opposite question: Did ethnic Jews considered themselves as Poles? In an article entitled, “Jews and Poles Lived Together for 800 Years But Were Not Integrated,” published in Forverts (New York, September 17, 1944), Yiddish author and Nobel laureate Isaac Bashevis Singer wrote:
Rarely did a Jew think it was necessary to learn Polish; rarely was a Jew interested in Polish history or Polish politics. … Even in the last few years it was still a rare occurrence that a Jew would speak Polish well. Out of three million Jews living in Poland, two-and-a-half million were not able to write a simple letter in Polish and they spoke [Polish] very poorly. There are hundreds of thousands of Jews in Poland to whom Polish was as unfamiliar as Turkish. The undersigned was connected with Poland for generations, but his father did not know more than two words in Polish. And it never even occurred to him that there was something amiss in that.
Is this true? It seems unlikely to me that the Jewish community could be so not integrated into Polish society, despite having lived there for hundreds of years. I would assume them to be somewhat insular, but not to the degree asserted in the source.
First of all, it is important to note that while Isaac Bashevis Singer is a real person who really grew up as a Jew in Poland, he is only one person, and cannot be a representative Jewish experience of Poland.
The first important thing to note is technical. It is impossible for a Jew in Poland to "not know more than two words in Polish", by definition. Polish Yiddish contained a huge vocabulary of Polish loanwords, even for basic terminology (even numerals, in some cases). Simply by knowing Polish Yiddish you would have a vocabulary in Polish, even if unable to read, write, or speak it. Yiddish has always accomodated loanwords from local vernaculars, while sometimes makes comprehension difficult between American and Israeli Yiddish speakers, or between Yiddish speakers from different areas of Eastern Europe (although the intra-Yiddish dialects can make things difficult there too).
The other important thing to know about Isaac Bashevis Singer is that his father (who he references as someone who didn't know any Polish at all) wasn't just some random guy, he was a Chassidic Rebbe. In such a position he would have no particular reason or desire to talk to the non-Jewish world. With Jews being a huge minority in Poland, and even a local majority in some areas, it would've been possible to live a normal life only speaking Yiddish. Which, it appears, Isaac Bashevis Singer's father was one.
However, that experience was not necessarily typical. A lot of Polish Jews did learn at least some Polish, in order to communicate with Poles for business. Even if you spent your entire social life in exclusively Jewish circles, it would still be useful to talk to Polish customers at your store (or talk to Polish shopkeepers when buying things), or talk to government officials, etc. Living in a Jewish social sphere doesn't imply zero interaction with the outside world.
But even then, not all Jews were living in a purely Jewish social sphere. A substantial minority of Jews were integrated to an extent into Polish society. In the 1931 census, about 10% of the Jewish population of Poland identified Polish as their primary language (2.5 million spoke Yiddish, which is the source of the numbers in your OP. The remainder mostly put down "Hebrew", which is exceedingly unlikely to be true, but I have not seen any information to clarify what those people were speaking or thinking when they put that down on the census). That does not account for bilingualism or trilingualism, of course.
The Jews who spent substantial time integrated into Polish society but still spoke Yiddish at home would not be measured in that census, but was sort-of measured in the 1921 census of Poland. As an aside, it is very difficult to tease out national identity from census questions, especially when they changed over time. Some censuses also grouped languages or ethnicities in certain ways for political ends, to exaggerate or diminish the perception of certain groups. The 1921 census of Poland did ask for "nationality". Of the 2.7 million Jews-by-religion in Poland, about 2 million identified their nationality as Jews, and 700,000 identified their nationality as Poles. While we can't know what those people were thinking (nor do I know the connotations of "nationality" in Polish, nor the way the question was phrased), it is likely that this represents a real substantial minority (about 25%) who thought of themselves as Jews but identified their nationality as Poles. This implies a high level of identification with Poles for that 25% of Jews.
Anyway, the details are somewhat hard to tease out for people not around to tell us, and even if they were, it might not be a simple answer. But a sub-group of Jews identified as speaking Polish primarily, and a larger minority identified as having Polish nationality. This means that, while the experience of one particular Polish Jew might be true for him, it is not representative of Polish Jews as a whole. Of course, most people were probably somewhere on a continuum in between, speaking some Polish, identifying with/socializing with Poles to some extent, etc.