I wouldn't go so far as to say that chicken is the closest living relative to the T-Rex, just that it is the closest in sequence similarity to this particular collagen protein, out of all of the protein sequences known in 2007 (chicken could have been the only bird represented in the database at the time, but I am not going to take the time to look into this).
I just searched the GVQGPPGPQGPR T-rex collagen sequence given in the text against the NCBI nr database, which is pretty comprehensive. It yielded the following Collagen alpha-1(I) chain perfect matches:
Brachylophosaurus canadensis [dinosaur]
Tyrannosaurus rex [dinosaur]
Sarcophilus harrisii [Tasmanian devil]
Monodelphis domestica [Gray short-tailed opossum]
Corvus brachyrhynchos [American crow]
Gallus gallus [Chicken]
Manacus vitellinus [Golden-collared manakin]
Pseudopodoces humilis [Ground tit]
Anas platyrhynchos [Mallard duck]
Geospiza fortis [Medium ground finch]
Acanthisitta chloris [Rifleman]
Columba livia [Rock dove]
Melopsittacus undulatus [Parakeet]
Falco peregrinus [Peregrine falcon]
Falco cherrug [Saker falcon]
So, that's 11 birds, 1 other dinosaur, and 2 mammals (one placental, one marsupial). The list gets bigger if we relax the sequence similarity cutoff. Based on this single fragment of a sequence, we can infer that T-rex is generally more closely related to birds than to mammals or lizards (and no lizards made the top-hit list), since there were a lot more bird matches than mammals (and the lack of mammal hits is likely not due to lack of sampling relative to birds). This is a big inference to make from a single fragment of a single protein, but I'm reasonably confident that further analysis of additional T-rex sequences would strengthen this finding.
If more sequences have been published since 2007, then perhaps we could get a better idea of which modern bird T-rex is most closely related to, but there is no way to determine this from just the single example sequence above. We cannot say with any confidence that T-rex is more related to chicken than to any other bird, unless a much more thorough analysis is performed using a lot more data. Perhaps this has already been done, but I haven't taken the time to hunt for additional literature.