I don't think this is true or at least is not reflected by any statistics.
Inequality, as measured by GINI has been falling both for the EU as a whole and within most European countries for decades now. The last decade in particular has seen declines in nearly all EU countries.
Taking the EU on average the last time it increased was in 2014 - which is to be expected as the 2013 expansion allowed a number of relatively poor countries into the union.
For individual countries, you can check the statistics here: https://data.worldbank.org/indicator/SI.POV.GINI
A few examples, I looked at Ireland in particular, and inequality has never been lower - the earliest statistics I could find is from the mid-1980s. Inequality peaked around 2000 for the UK and has declined hugely since then. France is more equal than it was in 2000 although the fall is less dramatic than that for the UK. Admittedly Germany has seen a slight rise since 2000 but Germany has also absorbed millions of very poor refugees in that time.
The US is an outlier globally - with rising inequality over the last few decades.
Slightly off-topic, this untrue claim of rising inequality in Europe is often presented without challenge and then used to justify some radical political solution. To my mind, it's using the same political "mind hack" that the the MAGA/alt-right in the USA have used (in this case concerning race, immigrants, global multilateralism, or social tolerance - "wokism" - in general).
For both, the veracity of the claims is apparently unimportant and uninteresting as long as the claim aligns with one's political orientation. The goal of these oft repeated untruths is to provoke indignation or anger - in order to drum up support for some radical political "solution".