In all my reading of Realms material over the years, I've never seen the “Roman” approach to other pantheons practiced or even referenced. Other pantheons are generally acknowledged as real, just foreign—a natural effect of living in a world where the individual gods regularly and undeniably manifest in person.
Instead, what you tend to see mentioned while reading Realms campaign materials are small shrines to individual gods or to pantheons as a whole. Where there are even a few worshippers of a god or pantheon, there will be a shrine somewhere where they can pay their devotions and tithe for its upkeep. Not all gods are represented publicly everywhere, since a public shrine to a hostile god would not last long facing the wrath of a local mob (and adventurers!); but such things tend to be either personal and private, or pop up in “underground” or black-market areas where they're overlooked—or better appreciated.
So if you're in human lands, you'll have to seek out some dwarven expatriates and find out where they worship their traditional gods. Failing that, you're on your own—you'll have to pray in this “wilderness” as best you can with your own devices, possibly constructing temporary, poor, personal shrines as you need to. After all, a paladin is a travelling sword of their god—it may be necessary to go to places forsaken or inaccessible to your god at times, as your god charges you to, and you can only do your best to honour them in such places.
In a magical medieval world like FR, I am not sure if it would be easy to find an institution of higher learning that is not connected to theology, military or magic. Even Earth's fair share of universities were founded with religious studies playing a core role.
As a concrete example, consider the so-called City of Splendors, Waterdeep, one of the largest cities in the continent of Faerun, and possibly the most influential. An entire 2e boxed set, and a 3.5e sourcebook was dedicated just to this city. The 3.5e sourcebook describes it to have over 130k population which increases 5-fold in the summer months. The city boasts 12 churches, 4 or 5 arcane schools, a large number of guilds, places to obtain poisons, potions, magical and mundane weapons, a great library associated with the church of Oghma, 11 independent sages (mostly mid-to-high level experts) specialising in various mundane and arcane topics, but only a single organization that is categorised as scholastic, the bardic college New Olamn. No universities at all.
If you decide to expand your question to allow temples and libraries in FR, then the list is a lot longer of course. The most famous of the libraries is Candlekeep. Another one would be the High Temple of Mystra on Mount Talath in Halruaa. Others can be found on the FR wikia. Furthermore, if you allow universities of magic, 3e FR Campaign Setting (FRCS) mentions that it is not uncommon to meet wizards who are university graduates in the lands of Lantan and Halruaa.
Finally, if you want to stick to universities and find the Zakharan and Oriental institutions listed in guildbounty's answer too far, you can try the Great University in Gheldaneth. To the extent that I know, that is the only university that is mentioned by name throughout the whole FRCS. (With its 320-pages full of two-column small-font text, FRCS is arguably the most information-dense sourcebook published on the Realms.) The Great University was one of the two most prominent buildings (the other is the Wizard College) of that metropolis, which happened to be the second largest city in Mulhorand, and ruled by clerics of Thoth. Unfortunately the city survived but got devastated by the Spellplague, so it is not clear what the current whereabouts of the university are.
Best Answer
There is the Cat Lord, the deity of the Tabaxi, and various animal gods from other pantheons, but as far as I'm aware, 5e doesn't specifically mention any canine gods.
Technically, it's not explicitly mentioned as being worshipped by anyone but Tabaxi, but presumably, the Cat Lord can be viewed as a deity worth worshipping by any feline species.
Since 5e doesn't include any canine species (aside from Gnolls maybe, but I wouldn't recommend Yeenoghu as a god to anyone), there doesn't seem to be an equivalent to the Cat Lord for canines. Thinking of blink dogs, powerful fey would probably make good patrons for any animal or canine humanoid species.
That being said, the official Forgotten Realms pantheon for 5e is on page 294, and it doesn't include a deity specifically dedicated to canines, felines or even animals altogether.
Suitable gods from pantheons aside from the default Forgotten Realms one include:
... mostly Nature-based gods, such as:
... and other gods that are not specifically listed as Forgotten Realms gods (see PHB Appendix B, starting p. 293). The (in my opinion) most suitable ones of those include:
Work with your DM to include the gods you want, such as Nobanion.
PHB p. 293 states:
The same page also states about the Forgotten Realms pantheon:
This explicitly describes the pantheon as being pretty extensive, rendering the list on p. 294 as a mere subset of the more popular gods, and making it very possible that Nobanion (and countless other gods) also exists in the 5e pantheon. However, the DM ultimately decides which gods exist in your DM's version of the Forgotten Realms, so just talk to them about your plans/ideas. Including another god doesn't really change the balance whatsoever, since you're not going to meet the god in person (ingame), and there are gods for any domain already anyways.
Adventurer's League
Side note: Adventurer's League (5e organized play) puts a restriction on the available deities, as described in the player's guide (link to current version):
This means that you're left with either the Cat Lord (which is mentioned in Volo's Guide to Monster, which in turn is "listed in Step 1") or with the nature gods from the regular FR pantheon. Your DM can't allow Nobanion or other "custom" gods in Adventurer's League.