The idea of incompatibility due to social status certainly comes up later in the story, in the context of the relationship between Polly Mooney and Bob Doran:
[Doran's] family would look down on her. First of all there was her disreputable father and then her mother's boarding house was beginning to get a certain fame. He had a notion that he was being had. He could imagine his friends talking of the affair and laughing. She was a little vulgar; some times she said "I seen" and "If I had've known." But what would grammar matter if he really loved her?
This is the main doubt in Doran's mind about marrying Polly, other than a more general dread at the whole concept of marriage. So it does seem to be a potential issue for him; however, the last sentence of the quoted passage above negates this to some extent.
But we also need to consider Mrs Mooney, Polly's mother. Throughout the story she is portrayed as an extremely capable and intelligent woman:
She was a woman who was quite able to keep things to herself: a determined woman. [...] She governed the house cunningly and firmly, knew when to give credit, when to be stern and when to let things pass. All the resident young men spoke of her as The Madam.
Her way of dealing with the affair between Doran and her daughter is a Machiavellian masterpiece. The strong implication is that she deliberately allowed it to go on long enough that he would feel obligated to marry Polly. She was aware of it for a long time (long enough that even the other lodgers became aware!) before doing anything about it, and she chose her moment carefully. She shrewdly manipulated events so that she could get her daughter safely married off to Doran.
There had been no open complicity between mother and daughter, no open understanding but, though people in the house began to talk of the affair, still Mrs. Mooney did not intervene. Polly began to grow a little strange in her manner and the young man was evidently perturbed. At last, when she judged it to be the right moment, Mrs. Mooney intervened. She dealt with moral problems as a cleaver deals with meat: and in this case she had made up her mind. [...] She had been made awkward by her not wishing to receive the news in too cavalier a fashion or to seem to have connived and Polly had been made awkward not merely because allusions of that kind always made her awkward but also because she did not wish it to be thought that in her wise innocence she had divined the intention behind her mother's tolerance. [...] she thought of some mothers she knew who could not get their daughters off their hands.
With that in mind, we can safely conclude that she would consider marriage to Doran to be a safe and healthy prospect for Polly. If she hadn't, she would've put a stop to their affair at an earlier stage, perhaps as soon as she was aware of it. It's a safe bet that if she's actively schemed to get her daughter married to this man, she doesn't consider them to be incompatible or an unhealthy match. Thus, for her, the difference in social status between them is not an issue.
And from that it follows that the reason for the failure of her own marriage wasn't, at least in her view, a difference in social status. If her marriage to Mr Mooney had taught her that such differences cause broken relationships, then she wouldn't now be trying to get her own daughter married to a man of a different class from herself. Thus the final conclusion is that Mr and Mrs Mooney's incompatibility was nothing to do with their difference in social status.
The only possible gap in this logic would be if Mrs Mooney is wrong about her own marriage - if it did fail due to their class difference but she believes there was some other reason for it. But given the consistent portrayal of Mrs Mooney as a highly intelligent and competent woman, this seems unlikely.
TL;DR: no, the Mooneys weren't incompatible just due to social status differences - because if they were, then Mrs Mooney wouldn't be manoeuvring her daughter into such a marriage.
