I have observed, from subtitled movies, that there seems to be an expression [tɔdʒe] in Cantonese meaning, approximately, "thank you". In Standard Lhasa Tibetan as represented by Nawang Nornang, the Tibetan expression meaning "thank you" is [tɔɔdʒe] (tones omitted, vowel qualities approximate). As far as I know, in Mandarin, "thank you" is [çieçie] (xiexie). This raises a few questions that I hope someone can answer. First, is it correct that there are significantly (phonologically) different expressions for "thank you" in Mandarin and Cantonese (我不會說中國話). All of my experience with Cantonese comes from movies. Second, if in fact Tibetan "thank you" is similar to the Cantonese version of "thank you" and not Mandarin, is there an explanation for why Tibetan would be more similar to Cantonese than to Mandarin.
There are two phrases in Cantonese meaning thank you and they are actually used differently. One is '多謝' and the other is '唔該'. Roughly speaking, '多謝' can be used when you receive gifts or help from others while '唔該' is used only when someone helps you but not when you receive gifts.
Concerning your second question, I don't think that you should conclude Tibetan is more similar to Cantonese than to Mandarin just because of one single example. Just like nearly all languages in the world have a phrase pronounced as 'ma ma' and means mother while you won't say that they are all similar.
Tibetan has no immediate relationship to Chinese, but there are many loan words. Most of these were introduced when Old Chinese was still dominant, namely when China became involved with Tibet from Tang and onward, with buddhist missionaries and so on.
Cantonese also retains a lot of features from Old Chinese. It's a matter of geography, Tibet and Guangxi/Guangdong being quite far away from the Mandarin epicenter in the Northeast. Wu also retains a lot of features from older Chinese, but was independantly separated from the northeastern influence quite some time ago, and has also been somewhat influenced by Japanese (again due to geography).
The first 10 Tibetan digits (1-10) are vaguely similar to Cantonese, and the Tibetan word for table if I remember correctly sounds similar to 桌子. The Tibetan and Chinese for "thank you" may sound superficially similar (and they don't to me) but the underlying words are completely unrelated. And unless you can read the alphabet or learn the system of transliteration you can pretty much forget about gleaning pronunciation information from things like "thugs rje che" ཐུགས་རྗེ་ཆེ་. And one can say Sino-Tibetan all day long... I prefer Tibeto-Burman, but it doesn't really get one anywhere.
As others have correctly pointed out, the Tibetan expression for 'thank you' is ཐུགས་རྗེ་ཆེ་ (Wylie: thugs rje che). The approximate phonetic pronunciation in the Lhasa dialect is thu-je-che (the 'gs' and 'r' being silent). Literally, ཐུགས་རྗེ་ (thu-je) means 'compassion' and ཆེ་ (che) comes from the word ཆེན་པོ་ (chen-po), which means 'great'. So when you are thanking someone, you are basically telling them that they have 'great compassion'.
Although it may sound like the Chinese phrase 多謝, the similar pronunciation is just a coincidence and the two expressions have nothing in common whatsoever.