I believe that it varies by country. The law provides protection for databases and data sets, where sufficient creativity/choice/curatorship has gone into its creation. It's separate from but similar to copyright law. So far it exists in several countries including the EU but I'm hazy whether it exists in the US (from memory it doesn't) and the exact criteria.
In any case such a website may not meet the criterion for copyright - although the original data creator (the stock exchange whose data is being used/reused) may have.
I don't know about IP law - is the fact of a current price (or dataset of such facts at different times) protectable as IP instead? Are facts generally? Can I declare facts such as my street address, or a number, as IP? Clearly commercial effort goes into it, and clearly it can be protected by contract but copyright/IP law is less clear to me. The question of copyright protection for a commercially valuable number has been tried in court, however, as has the question of a commercially valuable number which is constructed into a number that is notable for noncommercial reasons.
(As a possibly similar thought-question that might shed light on it - suppose a website is hosted using stock settings of some "off the shelf" website software, can they sue for copyright/look and feel over the appearance of their site? Probably not since they didn't put creative/novel/original effort into it. But the original software creator probably could.)
Hope this partial answer helps.
Updated to answer follow-up comment
First off, you can't force a person to do something based on a contract with someone else. Only the websites contract with the end user (terms of use etc) can do that. So the supplier's contract with the stock price website cannot in any way force an end user to do or not do anything. It can however require the vendor to set specific terms and enforce them. It can require the vendor to sue an abuser of the website terms (or allow the supplier to do so in their place and at their risk) if the supplier says to do so. Or it can allow for anything else agreed between the data supplier and the stocks website.
The flip side is that this would be a contract case for damages. Unless there are statutory damages (which don't exist in most countries and exist but might not apply in the US), the supplier would have to prove the harm and loss forseeably and directly suffered and would only be compensated for that alone. A lot of breach of contract cases just aren't worth taking to court for that reason - because you could prove the contract was breached but not that you lost anything, much less that your loss was a direct result and foreseeable by the end user.
(In the EU, you'd also have to be able to show that the terms were not unreasonable as the law protects consumers from gross disparity if ability to set contract terms, and fixed terms deemed unfair, but that probably wouldn't be an issue here).
The latter part of the followup is essentially discussing nondisclosure agreements. You don't have to own the data for that - there is legal consideration by each party and that's enough (each has given something to the other: "I will show you some information if you agree to these conditions"). But that's contractual too, and the same applies, if the user shared the data anyway or disclosed it, your damages would quite likely be limited to the loss you could prove, whatever the contract says.
Last the law on databases/data sets isn't something I know huge amounts about. Where such a law existed, a supplier would probably argue that sufficient original effort went into the collation and organisation of the data and the user would probably counterargue that this was mere storage of data and no original work was involved, or that any damage (if any) was negligible and would have to be proven. As databases have their own law and not pure copyright law, they might be right or wrong, depending on the jurisdiction and wording of the law.
Wikipedia articles of relevance to databases/datasets:
