If a PC has plate armor, Str 19, and dons a Cloak of the Manta Ray, can they swim 60 ft. as the DMG says (p. 159), or do they suffer penalties for swimming in heavy armor even with a magically-granted swim speed?
You can do whatever you like; what I would do is play it as written.
PHB p. 144
If the Armor table shows “Str 13” or “Str 15” in the Strength column for an armor type, the armor reduces the wearer’s speed by 10 feet unless the wearer has a Strength score equal to or higher than the listed score.
As it says, if they are strong enough they suffer no penalty to speed, if not then they lose 10 feet of speed.
PHB p. 182
Climbing , Swimming , and Crawling
While climbing or swimming, each foot of movement costs 1 extra foot (2 extra feet in difficult terrain), unless a creature has a climbing or swimming speed. ... Similarly, gaining any distance in rough water might require a successful Strength (Athletics) check.
As it says, a creature without a swim speed uses 2 feet of movement for every 1 foot covered (3 if difficult terrain). A creature with a swim speed moves 1 for 1 (2 for difficult terrain). As phrased, a swim speed does not exempt you from making a Strength (Athletics) check if required.
Significantly, it says nothing about the effects of armor on swimming.
Does this make a lot of sense? From a game perspective, yes - it gives simple rules that can be used with a minimum of complexity.
Can you rule that armor should make a difference? Of course, it says you can right on p. 6 of the PHB - if the PC says "I swim across in my Plate armor" then a perfectly sensible ruling is "You sink like a stone", however, given that this is not what the rules say, fairness dictates that you tell the player the consequence of their action before they undertake it and also before they make a commitment to spend in game money on an expensive anchor.
DMG p. 159
CLOAK OF THE MANTA RAY
Wondrous item, uncommon
While wearing this cloak with its hood up, you can breathe underwater, and you have a swimming speed of 60 feet. Pulling the hood up or down requires an action.
As it says, it gives you a swim speed - take that and apply it to the rules above.
"can they swim 60 ft as the DMG says (pg 159)"?
"are any of you DMs ruling that there are some penalties to swimming in heavy armor even with a swim speed?"
I cannot speak for "any DM", but by the rules in the book: There are no penalties to swimming in armor with or without a swim speed.
"Would you include a higher Str requirement with swimming?"
Me? No, by the rules in the book: The cloak gives you a swim speed of 60 feet. If you meet the strength requirement, there is no penalty to your speed; if you do not then your speed is reduced by 10 feet (i.e. to 50 feet).