Expression for someone who has fallen in love a lot

The term serial monogamist carries many of the connotations sought. The OED offers a brief compound entry for serial monogamist that references serial monogamy:

serial monogamy n. the practice of engaging in a succession of monogamous sexual relationships.

Unlike the terms mentioned by the OP, this term implies not just that the person (who doesn't have to be male) engages in sex with various partners but that the person engages in monogamous relationships with multiple people, offering more insinuation of love and romance than being a casanova.

The explicit implication that the person is monogamous could be seen as supporting the premise of "falling in love," though people who practice polyamory might protest that one can fall in love without being monogamous, which is true.

Perhaps the best argument in favor of using the term for this meaning comes from the popular mainstream associations with the phrase.

We all know that friend who hasn’t been single in, perhaps, her entire life. It might even be you, who knows? Typically that person doesn’t know that she’s a serial monogamist. Like Taylor Swift, she just thinks she falls in love easily. But we all know, it’s much more than that.

• Her Campus

The difficulties surrounding romantic relationships in modern society have popularized a version of monogamy, which may be termed "serial monogamy." In this version, commitment or exclusivity typical of monogamy is maintained but it is usually confined to a limited period. In this increasingly popular romantic pattern, people still believe in some moderate form of ideal love, but give up their basic pretense that it should last forever. The beloved is still regarded to be unique, but in many cases he is not so for the rest of our life.

• Psychology Today

A serial monogamist is a person who loves being in love, and always finds himself or herself in a relationship all the time!

• Love Panky