NO, but for the day you choose. And for a longer distance (forgot from how far in KM) also the morning after. It is clearly stated on the ticket as I remember-but in HU. Note: you also get a kind of receipt with Tax nr-this is not the actual ticket! If you struggle with the lingo-use the yellow machines in all main stations-these even give a small discount if you pay by card, but most also take cash.
BUT; for all IC-trains you need BOTH a REServation and the usual surcharge (based on distance again) for ´fast=GYORS trains. These will be less as 2GBP even for a long distance, but at busy times the rather short IC-trains (will have AC and wifi) will sell out a few hrs before. THESE are for the actual trains used-so that may mean a lunch break on the way doubles these surcharges, even though the base ticket remains valid and you have to preplan. Fast trains require the surcharge only-the main Szeged line trains have both Gyors (non AC ) and IC-cars in the same train.
Fares are indeed exactly based on the nr of KMs used-and this also means that in case of alternative routings the fare may differ a few HUF-the very same farescale now also applies to buses from Volan-it is planned to merge the 2 so as to better coördinate routes/timings. All in all its an extremely set up logical system-which kind of characterises many things in HU. It is just that you need some mental change to get the basic line.
There are no discounts for the occasional visitor from now out of EU GB, and even InterRail etc passes will hardly pay off. Seniors from 65 from EU ONLY!! get free travel on all trains/bus/tram/metro, but still have to pay the surcharges if needed.
There are a very few ride-at-will day tickets, but I think only in the summer season-for around Balatonlake and for the scenic area N-W of Budapest (Danube bend), incl some buses and even a ferry. I think those for BKK/BKV=BUdapest urban also includes train rides-on local (szemely, S) trains only in the city borders, but as I am EU-senior since some time, I do not have to deal with that anymore, so forgive me.