The Forgotten Realms is not "high magic"; it represents D&D's standard level of magic (DMG p.9, p23).
While magic is somewhat commonplace in the Forgotten Realms, this is really the baseline level of magic in Dungeons & Dragons 5th edition. The Forgotten Realms is defined in the rulebooks as a standard D&D campaign setting.
This is not to say that you can't run a campaign in the Forgotten Realms which is high-magic; the DMG explicitly tells DMs they may freely change the parameters of prewritten campaign settings. But by default, and canonically, 5e-era Faerûn is a world of standard magic.
What is the definition of a "high magic" campaign?
The term "high magic campaign" is not strictly defined in the D&D 5e core rulebooks, except to state that magic items are more readily available to player characters than standard (Dungeon Master's Guide p.38, Xanathar's Guide to Everything p.126). In a low magic campaign, by the same token, they are rarer than standard.
The standard campaign is the default D&D experience as described by the core rules. Dungeon Master's Guide p.9, "The Big Picture", defines the Forgotten Realms as one such campaign setting:
This book, the Player's Handbook, and the Monster Manual present the default assumptions for how the worlds of D&D work. Among the established settings of D&D, the Forgotten Realms, Greyhawk, Dragonlance, and Mystara don't stray very far from those assumptions. Settings such as Dark Sun, Eberron, Ravenloft, Spelljammer, and Planescape venture further from that baseline.
D&D's standard level of magic in particular is defined in Dungeon Master's Guide, p. 23, "Magic in Your World", making particular reference to the Forgotten Realms:
People everywhere know about magic, and most people see evidence of it at some point in their lives. ... In the city of Waterdeep in the Forgotten Realms setting, the Watchful Order of Magists and Protectors is a guild of wizards. These arcanists wish to make wizardry more accessible so the order's members can profit from selling their services.
Eberron is then cited as an example of a world where magic items are more common than standard:
Some D&D settings have more magic in them than others. ... in the world of Eberron, magic is as commonplace as any other commodity. ... spellcasters of low-level spells are common ... authorities and common folk are more likely to have access to and use the results of such spells. Buying commonplace magic isn't only possible, but also less expensive.
A formal definition of a "high magic" campaign can be found in the D&D 3.0 Dungeon Master's Guide (2000), p.164:
High Magic: Spellcasters and magic treasures are twice as common as presented in these rules, if not more so. Most characters have a level or two of wizard or sorcerer. Even a shopkeeper might be at least a 1st-level spellcaster. Magic items are bought and sold in clearly marked shops like any other commodity. Spells are used to light homes, keep people warm, and communicate. The function they serve is as commonplace as modern-day technology is in the real world.
By this definition, Eberron is probably high magic in terms of magic items, though author Keith Baker avoids this terminology, as he considers the ready availability of high-level magic, which Eberron lacks, to be a defining trait of high magic setting. We see here that there is some subjectivity in the term's definition.
The terms "high magic" and "low magic" were infrequently used in WotC D&D products after this point, and even the 3.5 DMG omits these definitions. In the preview article Warforged, Shifters, Changelings, and Kalashtar in Your D&D Game (2005), Stephen Schubert states that magically created humanoid life forms may be commonplace in a high-magic setting:
Because warforged are examples of magical creation at its peak, they're perfect for high-magic games. However, they could also be ancient relics of a bygone civilization that have recently been uncovered by adventurers.
In the Cityscape interview (2006), Ari Marmell describes high magic as something notably beyond the normal paradigm, though potentially local to a single place in an otherwise standard world:
Similarly, the book does indeed take some time to address certain high-magic or other unusual cities. (For instance, the notion of flying or planar cities is addressed in brief.)
Does the Forgotten Realms meet this definition of "high magic"?
No, not in general.
To begin with, the 5e Dungeon Master's Guide defines the Forgotten Realms as a standard D&D world, as mentioned above.
The Forgotten Realms Sword Coast Adventurer's Guide does not describe spellcasting classes as any more common than in baseline D&D. There are "a few" sorcerers in larger cities, and the are only slightly more common in some cultures (SCAG p.136); this concurs with the Player's Handbook which notes that sorcerers are rare. Wizards who advance in level find themselves with few peers.
Magic items are not described as any more common, cheaper, or readily available than standard D&D.
For more historic detail, the magic level of the Realms was canonically described in the D&D 3.0 Forgotten Realms Campaign Setting, p. 92, "Magic in Society":
Wielders of arcane magic—also known as the Art—are rare in most Heartlands societies. No more than one person in a hundred or so is likely to have any ability as a wizard or sorcerer.
The text goes on to describe that in the most concentrated places of magic, such as among high elves, the rate of arcane spellcasters is up to ten times higher, but that's still around 10% of the population, which is far below D&D 3.0's formal definition.
Even the World of Greyhawk, which sets the D&D 3e baseline for what is standard magic, has several high-level and epic spellcasters (the Circle of Eight, Mordenkainen, Gwydiesin of the Cranes) and several high-magic elements (the vanishing city of Rauxes, a shop in the Free City that straight-up sells any magic item, a nation-state run by a literal demigod, the Tomb of Horrors, Overking Ivid V's pit fiend ally freely casting wish, and numerous powerful artifacts).
The rarity of magic items, another defining factor in high-magic, is addressed on page 94 of the 3e Forgotten Realms Campaign Setting:
Magic is not technology. Wizards and clerics do not manufacture levitating elevators or mass-produce magic portals for simple convenience or crude commerce. ...
Nearly all Faerûnians, no matter how humble, have seen minor magic gimmicks at some point in their lives. Fewer actually own such treasures, but it's not unheard of for well-off merchants or low nobles to save their money for minor trinkets such as a pot that can make itself hot, or a broom that can sweep itself.
In a high-magic world, magic items are as commonplace as modern technology, so most people would have a magic cooking pot. In 3e-era Faerûn, they are rare enough that only the richest might own one.
Further, magic items are described in this book as primarily obtained by looting ancient sources, as well as produced at great expense by temples or the Red Wizards. This isn't the "bought and sold in clearly marked shops like any other commodity" of D&D's "high magic"; magic items are found or commissioned specially, and adventuring magic items are rare.
Magic of Faerûn p.136 says that hundreds of magic items are made each year; not quite the thousands or millions one might expect if it were as commonplace as modern day technology. Page 64 of that book notes that the open market is largely full of superstitious charms instead of real magic items, with the few real sellers operating secretively or on the black market:
Unfortunately, in open-market stalls, a spellcaster won't find the more special items. She's more likely to find herself rubbing elbows with young men and women seeking objects whose effects rely more on the power of suggestion and superstition than any real magic, such as a ring of love's eternal hope or a pendant of fertility.
Magic items aren't widely sold openly, in clearly marked shops; according to Magic of Faerûn p.65, Risa's Shop in Baldur's Gate is specifically unmarked, operates secretively, and is available by invite only. Shalush Myrkeer secretly operates a secret auction once per year in a warehouse upon the full moon.
Races of Faerûn does describe certain magic items as "common items" among certain races, but the definition of that term on page 5 is merely an item regionally available at a 10% discount due to it being more prevalent than usual. The average shield dwarf still can't afford to buy a +1 keen battleaxe; it costs 7,479 gp instead of 8,310 gp, but that's still beyond most NPC fighters' budgets, so it would be a misnomer to say these items are really common, and inaccurate to infer that magic items are generally common in the Forgotten Realms. It's not the same definition of "common item" used in 5e's Xanathar's Guide, which instead represents very minor effects like automatically-mending clothing or an orb that tells which way is north.
What's more, this was all 3e era—the canonical 4e and 5e Forgotten Realms transitional cataclysms are likely to have reduced the setting's magic even further since then. Notably, the Spellplague weakened some items and violently reduced the number of spellcasters:
Most permanent magic items, such as artifacts, were left intact at the end of the Spellplague but charged magical items were either destroyed, warped, or simply ceased to function. ...
The blue flames also infected portals and planar gates, spreading even further across Toril. Almost every part of Faerûn was affected by the Spellplague and certain areas were eliminated entirely, while others were created anew. Thousands of spellcasters were either destroyed or went insane due to the collapse of the Weave after Mystra's death. In Cormyr, a third of the War Wizards were either killed or driven mad. Those who survived lost their ability to use arcane magic.
The Forgotten Realms is a magical world with some exceptional elements, but it is not high-magic by the standards of Dungeons & Dragons. In D&D 5e this is even more true, since Forgotten Realms now increasingly defines the standard level of magic for Dungeons & Dragons, and historic changes have significantly reduced the amount of high-level magic in the world.