Are calculus and real analysis the same thing?
A first approximation is that real analysis is the rigorous version of calculus. You might think about the distinction as follows: engineers use calculus, but pure mathematicians use real analysis. The term "real analysis" also includes topics not of interest to engineers but of interest to pure mathematicians.
As is mentioned in the comments, this refers to a different meaning of the word "calculus," which simply means "a method of calculation."
This is imprecise. Linear algebra is essential to the study of multivariable calculus, but I wouldn't call it a calculus topic in and of itself. People who say this probably mean that it is a calculus-level topic.
In Eastern Europe (Poland, Russia) there is no difference between calculus and analysis (there is mathematical analysis of function of real/complex variable/s).
In my opinion this distinction is typical for Western countries to make the following difference:
calculus relies mainly on conducting "calculations" (algebraic transformations applied to function, derivation of theorems/concepts by methods of elementary mathematics, computations applied to specific problem)
analysis relies mainly on conducting "analysis" of properties of functions (derivation of theorems, proving theorems)
However, still this distinction is unnecessary:
(the most important) issues of "calculus" and "analysis" are very often linked together so that distinction is impossible (e.g. consideration of concept of limit in calculus due to Cauchy or Heine is actually the same as in analysis)
it makes artificial ambiguity in perception of mathematical analysis
it isolates common sense approach obtained from elementary mathematics and disables straightforward transition from elementary mathematics to higher mathematics
issues of "calculus" and "analysis" treated together enables acquisition of deeper understanding of subject by making extension from methods gained from elementary mathematics.
This is a purely anglo-saxon distinction. In most countries, however, there is no distinction between "rigorous" analysis and "non-rigorous" calculus. There are just different levels of analysis courses, e.g. "real analysis for engineers".
The term "calculus" itself just means "method of calculation". Even simple arithmetic is a kind of "calculus". What people in Anglo-Saxon countries refer to as "calculus" is actually just a short version of "infinitesimal calculus", the original ideas and concepts introduced by Leibniz and Newton. Nonetheless, even the lowest-level "calculus" courses usually refer to concepts that were introduced much later after Newton and Leibniz, e.g. Riemann sums (Riemann lived about 200 years after these two).