My institute requires us to publish our articles by ISI journals as the first priority, and Scopus as the second option. My question is: Is Web of Science the same as ISI?
Web of Science is a product. ISI was an organisation that used to produce that product.
The Institute of Scientific Information (ISI) was bought by Thomson Reuters in 1992.
Web of Science is now produced by Clarivate Analytics, which bought the ISI intellectual property off Thomson in 2016.
Thus, in answer to your question, ISI as an organisation no longer exists, but when it did, being indexed by ISI probably meant being in the ISI Master Journal List (or in the Journal citation Report - JCR), which is a service that is part of the Web of Science website, and is now produced by Clarivate rather than ISI.
Is Web of Science the same as ISI?
It depends on what you mean by ISI. If you mean "Institute for Scientific Information," then see the other answers here.
But ISI can also mean "International Scientific Indexing," which might look the same but isn't. A nice discussion can be found here, which says "the International Scientific Indexing should be treated with extreme caution and it is probably best just to ignore it as its impact factor has no meaning." (Many predatory journals claim to have an "ISI impact factor" but these are for the International Scientific Indexing, and not for the Web of Science.)