From the card itself, the Merchant gets the track data, which includes card number, expiration date, and cardholder name.
If the Merchant requires zip code verification, they'll get your zip code, obviously.
(Card-Not-Present Merchants often get address data for billing/shipping purposes, but you asked about physical stores... and they get that from the Customer, not the card itself.)
The Merchant can track purchases made with that card within their store(s), but not those made at other, unconnected stores. Be aware that sometimes multiple stores (e.g. HomeGoods, TJ Maxx) are actually the same "Merchant" (TJX Companies).
The Processor, on the other hand, can correlate a single card's activity across multiple Merchants. They don't generally have transaction details ("what you bought") but they do have amounts, categories, Merchants, times, all of which may be provided to the Card Brands (Visa, Mastercard, ...) upon request, or law enforcement upon a subpoena.
Each processor will have a different view. If Processor A handles Merchants A, B, and C, and Processor B handles merchants D, E, and F, then the Processors will have completely disjoint sets of data to work with. In general most Merchants use a single Processor; some load-balance across multiple Processors for redundancy and availability, but most transactions will only be seen by one Processor.
Processors do a lot of data analysis to provide value-add, but not to the extent of providing individual cardholder details across Merchants. Most such data analysis is done on large, anonymous buckets, but others, like householding, require identifying factors be used in the analysis.
Processors, Card Brands, and Banks can also make loose inferences about what you're buying based on the Merchant Category Code (MCC). These aren't very exact - those salted peanuts from the Exxon station might get classified as "Gas" - but they provide some guidance. These are the codes that Corporate-issued credit cards will use to block non-work transactions.
Finally, cards themselves are informative. Merchants can tell the difference between a prepaid card and a Black Card, and they can treat the cardholder differently in accordance with their status, for example extending discounts to higher-value-card holders. This is true not only in a physical store, where the Merchant sees your card; Processors can provide this sort of metadata to Card-Not-Present Merchants as well.
(The ability to determine the type of card is not unique to Processors; it's based on the BIN (the first 6 digits of the card) and you can look it up with freely available tools like binlist.net. However, since the list changes over time, and since it's only a portion of guidance, this is a service most usefully provided by a Processor. For example, anyone can tell if a card is a Black Card - but as a Merchant you might treat a Black Card with a high chargeback rate differently than the rest. Only the Processor can integrate that guidance.)