No. Other answers are pretty close, but miss important factor.
I won't repeat in detail what other say, just summarize that for SSH key+password to be multi-factor in your case, it would have to be "something you know" + "something you possess".
What I would argue is if you need only knowledge to effectively replicate "something you have" (so nobody can tell which is original and which is copy), then it is not "something you have" but "something you know" instead.
For example, if I can't remember my password and have written it on a piece of paper, it doesn't stop being "something I know" and become "something I have". It is still just password (even if hard-to-remember), and once someone learns it, they can impersonate me any time they want without me knowing. It is the same with SSH private key. It is just data, and data is by definition "something you (could) know (and effortlessly make an exact and indistinguishable copy of)".
The main feature for something to be "something I have" is how hard it is to copy by unathorized third party, as the main feature of effective "something I have" is that the only realistic way the attacker can have it is if I don't have it anymore (as I'm bound to notice I'm missing it).
Of course, there are many many grey areas, as mentioned in some posts. CHIP bank cards would be "something I have" today, as it is not possible (without a lot of effort, people and money) to make a authentic working duplicate. However Bank card authorized only by magstripe, which any cashier can make a copy of with $25 equipment and $1 of materials is no longer effective "something I have".
Also, as technology progresses, definitions change. Once upon a time, MD4 was cryptohash. Nowadays it is most definitely NOT - it is just a hash, no better at being a cryptohash than simple Checksum.
So, "SSH private key + passphrase" actually fails at being two-factor authentication method on two fronts:
- SSH private key is just information and not physical object, so it is by definition "something you know" and not "something you have".
- if some authentication factor is totally ineffective at making it harder for attacker to succeed in authentication, can it still be called an authentication factor? If your server enforces 1-character-maximum password length and no limit on number of tries, is it still authentication factor? In strict theory, it might be, but in practice it is just security theater.
Note that this does not mean that ssh private key + passphrase is bad: it is much better than plain password, or unprotected private key. But it is not 2-factor.
But if you want extra security provided by two-factor authentication in ssh, you can setup 2-factor authentication in ssh, preferably in addition to having it's private key protected with passphrase.