No, it's not grammatically correct. Love is a stative/state verb. It describes a state of mind and can not be used in continuous tenses (verb+ing). However, McDonald's campaign slogan for years was "I'm lovin' it!" (loving) It's not grammatically correct, but it's a catchy phrase that works in advertising. In reality, people are always inventing new ways to sound hip or cool. The language is constantly evolving, so there may come a time when grammar rules will permit stative verbs to be used in continuous tenses. These days, you may be able to get away with using phrases like this in informal conversation, but they would be considered incorrect on an English exam.

It's a contraction. English speakers will easily understand. The full sentence would be "I am loving the Mom life". In other words, I love the life of being a mother. There's a suggestion that "Mom Life" is a specific experience that women will probably share and relate with.