As per Wikipedia,
Biphenylene is quite stable both chemically and thermally, and behaves in many ways like a traditional polycyclic aromatic hydrocarbon. However, both the spectral and chemical properties show the influence of the central [4n] ring, leading to considerable interest in the system in terms of its degree of lessened aromaticity.
This implies that biphenylene is rather stable, despite its anti-aromaticity.
As for triphenylene, I'm not sure, and it doesn't say so anywhere, but perhaps the free energy of formation of triphenylene is much more, as it requires more molecules to be put together, leading to the reaction being non-spontaneous. However, benzyne does trimerize as well, forming triphenylene.
So benzyne can actually undergo both dimerization and trimerization, forming two different, but stable products.
As requested, a more primary source is available here. This site cites other sources, whose links I have provided in the quotation.
These lines say more or less the same thing:
Several recent publications (2 3 4 5 6) concluded that biphenylene possesses signiﬁcant strain energy of the order of 50–60 kcal/mol. These numbers are at least confusing since molecules that possess such tremendous strain can hardly exist. In the meantime, biphenylene is a stable molecule that has been synthesized a long time ago while its chemical properties show typical aromatic behaviour.