Saved searches
What a saved search stores, why it stores a query rather than a result set, and how it relates to alerts.
A saved search stores the query, not the records it matched.
That distinction is the whole design. A stored result set is a photograph: it stops being true the moment a record changes, and the reader has no way to tell when it stopped. A stored query is re-run, so what you see is what the catalogue holds now, with each record carrying its own verification date.
What is stored
The filters, the sort, and the name you gave it. Nothing about the results.
Why it matters for alerts
An alert is a saved search plus a subscription. Because the search is a query, the alert can compare two runs of it and report the difference — which is impossible over a stored list.
Limits
Saved-search allowances are per plan and are stated on the pricing page. Searching and reading every field of every record needs no account at all; what an account adds is keeping a search, being told when it changes, and tracking what you applied for.