I sometimes find it helpful to think of साथ as 'company' making मेरे साथ become '(in) my company'. The preposition implied from the oblique case.
I thought में = "I" and मेरे = "my"? Can मेरे also mean "me" (subject of the sentence)?
Here, मेरे comes along with the postposition 'साथ'.
The notes in the Activity skill explain it well. Quoting directly, "Instead of being simple monosyllables, some postpositions in Hindi use the format: के + [postposition]. For example, as just learnt, के पास. Other examples include के लिए (for), के ऊपर (above), के नीचे (below), etc.
Pronouns take the plural-object form for these postpositions. For example, मेरे पास, उसके पास, जूलिया के पास, etc."
In this sentence, मेरे साथ is the object of the phrase खेल रहा, therefore it takes the oblique case.
It didn't like "He and I are playing together." Not sure what that does not work.
The point is that मेरे साथ means 'with me/in my company'. "He and I are playing together." would be something like वह और मैं साथ में खेल रहे हैं।
Even in English there is a slight difference in connotation between the two.
I understand the use of मेरे with a postposition, but is it only the pronoun directly before the post position that takes that form? For instance could the sentence read “मैं उसके साथ खेल रहा है” ?
you'd have to change "है" to "हूँ", and it would change the literal meaning of the sentence to "i am playing with him/her". You convey the same sentiments though. I think it's valid.