It appears that Lenin did indeed own several cats over the years.
According to Louis Fischer's The Life of Lenin (the source cited by the Wikipedia article):
He had had a brown cat in Geneva. During her visit to Lenin at Gorky shortly following the August 30, 1918, attempt on his life, Angelica Balabanoff saw two cats in the household. Lincoln Eyre, an American journalist who went to Lenin's Kremlin apartment, noted that the dictator was "fond" of his several cats.
One reasonably well-known image that you may have seen is this one:
(click to enlarge)
According to this article in the Guardian newspaper, the photograph was taken in 1922 by Lenin's sister "at his residence in the village of Gorki", and so could very well show one of the two cats seen by Angelica Balabanoff in 1918.