There appear to be no actual mechanics for magebred animals in 5e
A search on dndbeyond.com for any of the terms "magebred", "magebreed" or "magebreeding" should find any officially published 5e material which uses the term, but it only turns up the same references you have already found - I can't find any game statistics for magebred animals anywhere. It appears that there is no official 5th edition resource which provides game rules for magebred animals.
Widening the net slightly, Keith Baker (Eberron's original creator) has published Exploring Eberron on DMsguild. This book also references magebreeding several times and even includes some options for player characters from magebred backgrounds, but unfortunately does not include any stats or rules for magebred animals.
Unless and until any future Eberron sourcebooks feature such rules, this means that if you want to include magebred creatures in your campaign, their stats are up to you to create. It's not a terrible idea to reference the 3.5e magebred animal template from the Eberron Campaign Setting to do so! You have noticed that 5e is not as template-heavy as previous editions, but they still aren't a completely alien concept to the system - the Monster Manual does actually include 3 different templates, for use in creating half-dragon creatures, dracoliches, and shadow dragons, so there is precedent. However, they do mostly presume you will rely on the rules for modifying or creating new monsters in the DMG to figure out the resulting challenge rating.
Using the 3.5e template
If you do decide to refer to the 3.5e template to create magebred animals in your game, I don't think it's too tricky to adapt, and even calculating the challenge rating adjustment actually works out reasonably easily on paper; a +2 to AC would improve the creature's defensive CR by 1, and a +4 to the ability score used for attacks causes a +2 improvement in attack bonus, which also increases offensive CR by 1; by the DMG's rules, you'd expect those changes to increase the creature's CR by 1 overall. (Note that for creatures with very low CR to begin with, my previous experience analysing CR calculation suggests that fractional CRs may be better "stepped" along the table rather than increased by full points - e.g. for a CR ¼ creature, a 1 point increase in CR should probably map to CR ½, not "CR 1¼".)
The other changes the template can make aren't relevant to CR calculations, but care should be taken about what it says about Intelligence; in 3e/3.5e, animals could never have an Intelligence score higher than 2, so that's the smartest they could possibly be. 5e doesn't have the same limitation, and many animal-type creatures have better Intelligence than that already, so I would probably just change that to a +2 Intelligence increase.
The ECS also states that a magebred animal should cost twice as much as a normal animal of its kind, which seems a reasonable rule to use when pricing these magebred creations in your game.