Is there a theory of universal meaning?

First, you need to understand the position of 'Universal Grammar' in linguistics. It is a particular theory that grew out of very specific concerns - mostly having to do with learnability of syntactic structures. It posits a set of universal innate constraints on the syntax of languages - and is not really a grammar. It is highly controversial and linguists don't generally refer to it to solve problems of comparison unless they work within the theory.

While there is no 'Universal Semantics' that parallels Universal Grammar (in name or in theory structure), concerns with meaning universals go as far back as Plato (in the Western tradition). When Aristotle was outlining his principles of logic, he was in fact thinking of them as semantic universals. In this tradition, many people have dealt with formal logic and logical semantics as if they were in fact describing universal principles.

Cognitive semantics, on the other hand, posits universals in the shared human experience of the world. We all walk upright, have a similar field of vision, interact with objects in our environment, are born to women, grown up in a social setting, have similar internal biologies, etc. These are all reflected in how language is structured both at the level of syntax and meaning.

(Both the formal and cognitive approaches are described by Lakoff in Women, Fire and Dangerous Things).

Some people have sought to go even further and posit universal semantic primes (most notably Anna Wierzbicka).