When people took (or received) the last name Rechnitzer for the first time, they most likely did not live in Rechnitz anymore. They probably lived somewhere else, but perhaps retained a memory that (one of) their ancestors had been from Rechnitz.
The name Rechnitzer existed in quite a few places of old Hungary.
The name is not Jewish per se, but many Rechnitzers were Jews. The town of Rechnitz had a very strong Jewish presence, about 1,000 strong (and 20% of the total population) in the mid 19th century.
You could look through the residency records of Vienna (Meldezettel) which cover about the time from 1885 till 1925. There is a male and a female series of these residency records. Since I searched for the name Rechnitzer in Vienna about 1 1/2 years ago, I know that LDS film 1706876 definitely contains the entire male Rechnitzer sequence of residency records (spanning about the middle to 2/3rds of this film). Perhaps you will find a trace of your ancestor Albert when he visited (if he did) his kin in Vienna. In old Austria (and still in today's Austria too), there was an obligation to register with the police whether you lived in a town of visited it.
There are also residency records of Gross-Wien filmed by LDS. They contain very rich personal data and cover the time from 1930 till 1947 (including deportations of Jews). However, since you do not know the first names of Albert's relatives, you might have a hard time in assessing the relevance of individual records there.