I recognise that "Ma" is sometimes, though rarely, as far as I can see, used as a familiar for of "Mary", but even if that is involved here (and personally I tend to doubt it), I am fairly sure that the primary meaning is "call me mother". It's true that in English it could be taken either way, but I expect you will have to use a translation that cannot be taken either way (that is, you will not be able to use something that could be either a familiar word for "mother" or a familiar form of "Mary"), and that you will have to retain only one of these meanings. I am almost certain that you should retain "mother".

I think that in this situation, Sister Mary is being explicitly maternal towards one child and not towards the other, and that that is the key point. It's a more potent point than that one should be allowed to be on first-name terms with her and not the other. And it does not seem very likely to me that a nun would be inviting a small child to call her by her first name.

These blog critics of the series clearly takes it unambiguously as "mother", not "Mary", which doesn't prove anything, but at least shows the spontaneous reaction of a couple of keen viewers:

"Also, it’s fun to get some background context: that if Sister Mary was Lenny’s “Ma” (and let’s unpack THAT in a second), then Cardinal Spencer was Lenny’s Pa. [...]
What’s up with Andrew Dussolier, another child from Sister Mary’s orphanage that grew up to be a Cardinal? I believe we get a flashback to Andrew arriving at the orphanage, where Sister Mary tells him “Never call me Sister Mary, call me mom,” which is the exact opposite of what she told baby Lenny when he arrived."
http://observer.com/2017/01/young-pope-recap-episode-two-kan...

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Note added at 7 hrs (2017-02-07 10:00:50 GMT)
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I didn't explicitly answer your question 2. I'm afraid I have no personal experience of orphanages run by nuns, so I don't know whether it's common for orphans to call nuns "Ma (mother)", but I don't find it implausible that a nun would recognise the importance to a child of a mother figure and want to play that role. It seems likely to me that some nuns, having given up the possibility of having children of their own, might be glad to have a way of expressing their maternal instincts.

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Note added at 12 hrs (2017-02-07 15:41:37 GMT)
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To answer your first question directly: "Ma" as a short form of "Mary" is extremely rare. I can find one example on the Internet apart from the one JW Narins has quoted. When pronounced "Mar" I don't think it exists at all. As Tony says, people called Mary are sometimes called "Mare", rhyming with "where", but I have never seen this written as "Ma", though that's probably how it's meant to be pronounced in the couple of examples we've found. The name Mary is not usually abbreviated at all. Molly, Polly, Mae and May were all originally familiar forms of Mary, but nowadays people regard them as separate names.

In any case, even assuming it is plausible that Sister Mary would not only invite a child to call her by her first name (which I find unlikely) but by a familiar form of her first name, it is exceedingly unlikely, in my view, that it would be "Ma". I don't think "Ma" is even ambigious here; I don't think it is meant to be a form of "Mary" at all. I think its only meaning is "mother".