I've not used Monday, and I only used Asana for a few days when it first came out years ago. But i have used Monica somewhat minimally over the last year. So, my comments don't offer a fair comparison...But what i can say is that Monica hits the sweet spot of more than someone manually keeping notes in their mailbox or addressbok....but less than something like Salesforce, etc. Its actually well built from its minimalist UI to its succinct feature set to its basic (but good) notifications emails. If you're a sales person, or some actor's agent, or maybe work on a sales team, then maybe a more comprehensive CRM might be suitable...but for most other use cases, something like Monica works, and works quite well. They also offer a free plan (https://www.monicahq.com/pricing) with feature and contact limits, but its more than enough to get a feel for the service; to see if you might like it or not. (Considering that this is a side project for the creators, the limits are fair in my opinion.)
I'm going to split hairs here, but i disagree that it is only for specialized purposes. I honestly believe that it is good for general purpose, but simply smaller scale. So, yes, agreed with your note on it being smaller scale. But i do believe that it can successfully be used for many, many scenarios fit for a CRM. I stick to my opinion that anyone thinking that they need functionality of a CRM should begin with a smaller scale tool like Monica, and then over time see if an "upgrade" to a larger-scale CRM is appropriate. Just my opinion of course. :-)
I haven’t used Monica yet (though I’ve been considering it for a while), but I can tell you straight away that Monday and Asana don’t let you self-host.