I’d agree that the sentence could be interpreted in two somewhat different ways. However in either case, the monkey hugged the tiger.
A: It was the tiger that the monkey hugged (and not some other tiger).
B: It was the *tiger* (and not some other animal) that the monkey hugged.
Explanation: English word order is stricter than Russian, since we don’t have grammatical cases the way Russian does. To identify the subject of a relative clause (“that the monkey hugged”), simply look at the noun immediately before the verb — that is always the subject. No exceptions. This is true even in questions within a relative clause: “I want to know what animal the monkey hugged.” In our example, “that” is the object of the relative clause. If “that” were the subject, it would come immediately before the verb: “It was the tiger that hugged the monkey.”