On the one hand, yes. Premarital sex was considered shameful among middle- and upper-class society in the United States well into the twentieth century; contraceptive devices were considered "family planning", to be used by married women who wanted to limit the number of children they had, and it even by the end of the 1960s many unmarried women kept their use of the birth control pill a secret because it was often considered a means to enable promiscuity when used outside of marriage. Having an illegitimate child was another level or two more shameful, since it was indelible proof of her premarital activity, and unless she could be hidden somewhere far from her acquaintances and give up the baby to someone else, she would probably never get married, which would - at this level of society - probably mean living with her parents until they died.

You'll note that I was careful to say that this was the case only among middle- and upper-class society - because the working classes were often much more understanding about these things. Couples would often live together without being married, have children, and split up somewhat casually, since they didn't need to procure a divorce. There was much less shame attached to premarital sex and illegitimate children, simply because they happened more frequently as a part of life. And this also plays into the scene in the film, because Lady, Jock, and Trusty represent the upper middle class who would find illegitimate puppies earth-shattering, while the Tramp would have been easily supposed to have "loose morals", take advantage of a nicely-brought up young dog, and abandon her with a litter.

You can read some more statistics about this issue in the eighteenth and early nineteenth century in a previous answer of mine here.

On the other hand, I think the inclusion of this scene was just as much about Victorianism, which I discussed in this previous answer, as it was about accurately representing the 1910s. By 1955, there was more publicly-voiced anxiety about promiscuity and illegitimacy, and a significant amount of nostalgia for the supposedly more upright and moral Victorian and Edwardian periods (mirroring the way we often use the 1950s today), and so it serves to make the adult viewer go, "Wow! Look at the way these good male dogs are standing up to take care of their friend. You wouldn't see that today." Jock and Trusty also are both based on ethnic/cultural stereotypes - Jock is a Scottish scottie dog, and Trusty is a southern bloodhound (which is seriously creepy when you think about historic uses of bloodhounds in the American south) - and these stereotypes both include a focus on "honor". It would be honorable to marry a woman you aren't in love with and know not to be in love with you to save her from social ruin, so it was appropriate to have the situation come up and these two stereotypes to take that position.