Well it doesn't get any happier than a singing bird, does it? This simile is also appropriate because, in a way, the speaker's also singing about her own happiness in the lines of the poem.
My heart is like an apple-tree Whose boughs are bent with thickset fruit; (3-4)
The biblical association with the apple has it linked to the forbidden fruit, one that Eve was tempted into tasting—which resulted in her and Adam's expulsion from the Garden of Eden. This description seems pretty Eden-like to us, though.
My heart is like a rainbow shell That paddles in a halcyon sea; (5-6)
The allusion here is not to Christian religion, but to Greek mythology. The halcyon bird (a.k.a. the kingfisher) is a colorful bird that made its nest on the calm sea, according to legend. In other words, everything is nice and peaceful for our speaker.
My heart is gladder than all these (7)
What's happier than happy, gladder than glad, more peaceful than peaceful? We're not really sure, but the speaker's sure experiencing it.
Because the birthday of my life Is come, my love is come to me. (15-16)
These concluding lines let us know that the speaker's happiness is rooted in a profound appreciation for her life. She's not simply in a good mood; her entire existence is validated. That's some intense happiness.