I've done a video on it as to how it can effect it. I've included the link in my signature. I tested the theories with one seeming to reveal that perhaps it does? Try it yourself....
Extensive research was already done on this since the game was released in the summer. Dodge windows, quick and charge move durations, etc. are all exactly the same regardless of Pokémon size and weight. Every move in the game has a specific cooldown and duration. All dodges can be performed with the same window of time, with the only outside factors being player timing and server lag.
Size and weight also have zero correlation with Pokémon IVs and base stats, with the latter being exactly the same for Pokémon of the same species. A Chansey will ALWAYS have higher base HP (or in Pokémon GO, stamina) than any other Pokémon in the game. An Alakazam will ALWAYS have higher attack stats than a Clefable. These stats are formulated from those pulled directly from the core series games.
And FYI, Pokémon can even go from XS to XL and vice versa just through evolution. Size/weight have nothing to do with their stats. Here's an immediate example just from two Metapod I pulled up:
The first Metapod started off as a XS Caterpie and became a standard size Metapod (not XS. Not XL). It has more HP than the second "XL" Metapod, despite the fact that the second Metapod is at level 21 while the first is level 20. Why the difference? Because the first Metapod has a stamina IV of 14 while the second Metapod's stamina IV is 7. That's all there is to it.
More examples:
Two Pichu. Both level 20. XS one has more HP than the XL one. Why? Because the XS one has a stamina IV of 14 while the XL one has a stamina IV of 12.
More examples:
The first Dragonite isn't marked as XS or XL, but you would say 255.93kg is a bit more than 210kg, right? Then why does the smaller and lighter Dragonite have more HP? Both are level 32. Because the first Dragonite has a stamina IV of 11 and the second one has a stamina IV of 15. That's all there is to it.
For more information on how CP, IVs, and levels work, see here:
https://www.pokecommunity.com/threads/377501 (the only outdated information is the actual CP formula, but the basis is the same).
As mentioned already, the sizes are gimmicks and the only things that are impacted by them are the two medals. If you want references, the Youngster medal is for Youngster Joey from Gen. II made famous from his "Top Percentage Rattata" meme. As for the Fisherman medal, it's also a reference to Gen. II, specifically the fisherman at the Lake of Rage who dreams of catching the biggest Magikarp and measures any of those you catch.
There's not much room for theory here considering this is literally how the game works.