Agreed you should raise more on the turn. Because he donked small, you need to make a bigger relative raise to lower his pot odds. Here, he's asked to call .28 into a pot holding .86 i.e. over 3:1 pot odds. He's priced out if he has a flush draw, but if he's going to call .28 there, he's very likely to call somewhat more. Plus you set up your stack to shove around the pot on the river.

As for what to make of his donking both the turn and river, the turn is a blank unless he has 44, which is fine with you anyway. So what else might he have to call oop pre- and then to check-call the flop before donking small on the turn? Maybe a flush draw hoping to see the river cheaply, but since he might fold at least some of them, it seems more likely after that he calls your raise that there are more combos in his range that are decent made hands.

When he donks again on the river, I tend to think he's not playing you for the flush. This isn't a strong feeling, but being on the passive side, he'd more likely tend to assume you wouldn't play a draw this way because he probably wouldn't.

I'd guess he has something like QJ or 88. I don't think either calling or raising is awful, but I'd probably call, in part because I didn't price out his draws more on the turn.