It does not grant extra benefits
The grappling rules do not say anything about size other than that you cannot grapple something two or more sizes larger than you, and that your speed is halved during a grapple unless the grappled creature is two or more sizes smaller than you.
By RAW, no advantage due to size is granted for any grapple checks.
The Enlarge/Reduce spell has multiple effects. One is to enlarge or reduce the size of the creature. Another is to grant advantage on Strength checks and saves. That is, the increase in size does not cause the advantage to Strength checks and saves. Both effects are caused by the magic of the spell.
In the spell description, notice how the advantage benefit is written. It is said as "also, you have this benefit" and not "because of that, you have this benefit" -- suggesting non-causally related items in a list.
Until the spell ends, the target also has advantage on Strength checks and Strength saving throws.
If it was the size change that conferred the benefits of advantage, then a creature who was Medium size and enlarged to Large size, when grappling a creature of Large size, should not get advantage on their checks. But they do, because magic, literally.
Bigger monsters typically have higher Strength scores
In a way, being bigger does have advantages for the sake of grappling. But this is reflected in their high Strength scores and proficiency in Athletics check (if they have it). It is not reflected by any application of the advantage/disadvantage mechanic for reasons owing purely to size.
Tentacle Slam is not technically an attack.
In the stat block, the other actions include the phrase Melee Weapon Attack while Tentacle Slam has no similar verbiage. Tentacle Slam does not call for making an attack roll with a d20.
Tentacle Slam is an action that causes grappled creatures to make saving throws. It does not qualify as an attack for the purposes of the Otyugh's Multiattack ability.
When an attack is not an attack
This prior answer explains what counts as an attack in the game mechanics. That answer draws upon the passage from the Player's Handbook:
If there's ever any question whether something you're doing counts as
an attack, the rule is simple: if you're making an attack roll, you're
making an attack (PH, p. 194)
There are many actions that would likely be considered "attacks" in-universe, but which are not attacks mechanically. Examples include grappling and spells with save versus damage effects.
When reading a term in the rules, interpret it with the definitions in the rules. When deciding how NPCs/monsters react to something, use common sense.
Don't confuse the Grappled condition with the "grappling" attack option.
When you want to grab a creature and hold on to them, the "grappling" attack option is how you accomplish that. The rules for doing this include a size limit on the target. If you succeed, you apply the grappled condition to the target.
But the size limit is a function of the grappling attack, not a rule that applies generally to the grappled condition. A condition is what it is, as defined in the rules appendix A (PHB p.290).
Spells, feats, magic items, or monster special actions may provide other ways to inflict the grappled condition without using the "grapple attack" rules, and therefore have no inherent size limits; the spell, feat, item, or action will specify what limits, if any, apply to it.
So yes, in theory, a giant frog could grapple an ancient dragon and stop it in its tracks. As a DM, you should feel free to overrule this if you feel it's absurd, but it's technically what the rules say.