How many loanwords in Chinese have Chinese equivalents?
Example: 三文鱼 is a loanword, borrowed from English, meaning salmon, but Chinese also has the word 鲑鱼 meaning salmon.
Are there a lot of these kinds of word-sets?
I'm not sure where you could get an accurate count for how many there are. Considering that loanwords have been coming into Chinese for thousands of years, it definitely won't be a trivial task.
There is certainly quite a few, however, not all of which is current/widespread/universal. I'll list some here, and edit more in if I think of any later:
奶酪 / 乳酪
發動機 / 動力機 / 機
索引 / 檢目 / 總檢
舞會 / 聚會 / 聯歡會
蟲膠清漆 / 酒精清漆
鰛 / 鰯
計程車 / 出租車
Adding to the previous list:
Bowling: 保龄球 bao ling qiu
Buffet: 蒲飞 pu fei
Calorie: 卡路里 ka lu li
Cartoon: 卡通 ka tong
Motor: 摩托 mo tuo
Sundae: 新低 xin di
T-Shirt: T-血 T-xue
Toast: 多士,吐司 duo shi, tu si
Vitamin: 维他命 wei ta ming
During the May Fourth Movement, many terms were "imported" from Japan to enrich the Chinese vocabulary for translation of Western idea. Not to mention that China, Hong Kong and Taiwan have been different translation for the same English word, for example:
Cheese = 芝士 (HK) / 起司 (Taiwan) / 奶酪 (China)
Toast = 多士 (HK) / 吐司 (Taiwan) / 烤面包 (China)
Hence, it is very difficult to come up with a definite answer to your question.
If I read it right, @user3306356's question is whether there are a lot of word-sets comprising a word that has long existed in Chinese and another word that is a loanword from another language representing the same concept.
In this sense, I will say there are not a lot of these kinds of word-sets. Anyway, if there is already a native Chinese word for a certain concept, why bother borrowing a new one from another language?
A loanword is a word adopted from one language (the donor language) and incorporated into a different, recipient language without translation. That is, only the pronunciation of the word is somewhat reserved, and you can't reason about its meaning from the Chinese characters makeing up the word.
Sofa = 沙发 is of this kind.
A calque or loan translation is a word or phrase borrowed from another language by literal, word-for-word, or root-for-root translation.
E-mail = 电子邮件 and
skyscraper = 摩天楼 are of this kind.
Chinese seldom borrow new words from other languages for something already existed in Chinese. There are a few word-sets of this kind, and the loanwords are used just because they sound fancy or have an exotic feeling.
Bacon: 培根 = 煙肉, Daddy: 爹地 = 爸爸, Cheese: 芝士 / 起司 / 起士 = 奶酪 / 乳酪 are of this kind. The Chinese words before the equals sign are loanwords (phonetic translation).
When new concepts are introduced to China, it's most common for Chinese to use existing Chinese morphemes to coin new words for the concepts. Chinese love free translation and phono-semantic matching much more than phonetic translation. Anyway, a word made up of native Chinese morphemes is much more comprehensible than a word made up of some characters having nothing to do with its meaning. At first, a phonetic translation and a free translation or phono-semantic matching may exist at the same time and compete with each other, and usually the free translation or phono-semantic version will beat the phonetic translation at last.
Many word-sets listed by @Semaphore are just different Chinese translations of the same new concept imported from other languages, not word-sets made up of a loanword and a native Chinese word for a certain concept that has long existed in Chinese.
Laser: 雷射 = 激光 as an example. Laser is invented in the USA, so it's a new concept from English. Only the concept of laser was borrowed, not the word itself. 雷射 is a phono-semantic matching of laser, 激光 is a free translation of the same word. But neither the word 雷射 nor 激光 existed in Chinese before laser was invented. So this word-set doesn't conform to @user3306356's standard.
Taxi: 的士 = 計程車 / 出租車 is similar to
Laser: 雷射 = 激光. 的士 is a phonetic translation of taxi and is mostly used in Hong Kong. 計程車 is a free translation of taxi and is used in Taiwan. 出租車 is another free translation of taxi and is used in mainland China. The concept of taxi is imported from western world. 的士, 計程車, 出租車 are just different translations of taxi, none of these words existed in Chinese language before the concept of taxi was introduced to China. So this word-set doesn't conform to @user3306356's standard either.
Again, I couldn't comment due to not enough reputation so I'm posting an answer here.
Romantic: 羅曼蒂克 = 浪漫
is incorrect. they are both translated by sound.
It is said a Japanese (actually a very famous one), Shouseki Natsume (夏目漱石), created the Sino-Japanese word 浪漫(ロマン) from the first moras of romantic（ロマンチック）. While who did it is still debated, it is certain this is a loan word that is translated by its sound then clipped. In Japanese, ロマン is pronounced almost identical to Chinese 罗曼.
I must also comment: This question is asking for an endless list. What's the point? Can I create a question that ask for every Chinese characters ever existed?