My interpretation is thus;
He asks Tifa because they are alone (Tifa asked him to keep the Zack information between themselves) and he just got caught staring at her with a smile on his face, it was a classic case of just changing the subject due to embarassment. He has already ascertained that Aerith might still have feelings for Zack due to her response in Gongaga and he was rather rude towards both Zack and Aerith when he was trying to be supportive. Why do i think he was trying to be supportive and not jealous? Well, look at the very intentional close up of his face, it's a stone cold neutral face with virtually no emotion.. not only that but when he calls Zack a loser he looks to Tifa for reassurance/backup that what he is saying is the right thing to say to support a friend. He obviously comes to the conclusion that he overstepped the mark (more on that with the Aerith date) and you can see the POV camera following Tifa as he claims "another time" obviously to talk to Tifa about this situation later on. Obviously he gets even more of a conclusion in Nibelheim where he remembers not only who Zack is (his friend) but that Zack also was extremely in love with Aerith, so now he's in a positions where he knows the feelings of two of his friends but doesn't know how to approach the subject. Tifa offered to talk and once she says no she hasn't yet he comes to the conclusion Tifa is finding it hard to approach the subject with Aerith because it's a touchy subject due to what he believes to be Aerith's lingering feelings for Zack. Notice the camera is not on Cloud's face this time and his line delivery is pretty neutral, this is because we've already established that Cloud is not "jealous" of the situation and is instead mearly concerned for Aerith. Contrast this moment, for example, with Yuffie telling Cloud that Tifa forgot about him when she came to Midgar, when the camera switches to a close up of Cloud's face as he shows a visible distate for Yuffie's comment because the dev's intention is to show us Cloud's emotions in the moment.
Now lets talk about the Aerith date - I'm not sure if his intention was to bring up talking to Tifa at any point, maybe it wasn't. But notice the two very different situations that bring about the conversation, in the Aerith date this is after she basically confirms to Cloud that her initial draw to Cloud was that she reminded him completely of Zack, and you can tell not only by Cloud's facial expression but by the way Cody delivers the line that he's slightly uncomfortable with the topic because he "remembers" Zack and his feelings for Aerith, as well as what he believes to be Aerith's lingering feelings for Zack. He wants to know if Aerith knows the same thing, but she says she doesn't, and his "Oh.." whilst turning away kind of leads him into a slightly more uncomfortable situation. because she's saying these things to Cloud whilst he has to deal with the knowledge that he believes Zack is dead, and that his best friend was in love with the girl who he's in a rather intimate situation with currently. Notice he doesn't mention Aerith's feelings for Zack because at this point he remembers and has learned from Gongaga, just as he says on the Tifa date he recognises it's a "touchy subject" because of her lingering feelings for Zack and he doesn't want to voice them infront of Aerith incase it upsets her. The two situations mirror each other. In this moment I think Aerith understands or clocks on to Cloud's apprehension and sudden anxiety and that's why she does what she does best and diffuses the situation somewhat by latching onto his arm and asking Cloud if it's okay to just pretend "until the ride is over." Now I know a lot of people will see the handhold as romantic, and if we're being honest, it can have romantic undertones for sure, but if we look at the context of the situation that just happened, Cloud's confused as to why Aerith just latches onto his arm, and understands the whole situation of the "date" just changed after the awkward Zack exchange, Cloud chooses to show us that he has emotionally matured since Gongaga and offers Aerith comfort that quite frankly she seems to seek out in the moment, hence "Until the rides over" and that's how the date ends.
I think the scene was important in both dates because it establishes that both Tifa and Aerith are keeping knowledge from Cloud because of the very complicated nature of the situation between Cloud, Tifa, Aerith and Zack, whether you want to believe this is purely romantic based, or Cloud's false remembering based, well it's up to interpretation, but certainly Tifa and Aerith are talking about a lot of things behind Cloud's back and things we certainly don't see on screen.
Notice also the difference in the climaxes of both dates;
The Kiss happens way after the Zack conversation, infact they talk about themselves first before anybody initiates anything
The Handhold happens almost immediately after the Zack conversation when Aerith latches onto his arm to break the tension of talking about Zack and his connection to Cloud, and how Aerith see's Zack in him. Even though, yes, she suggests that it's okay that Cloud isn't Zack and that she's currently trying to find Cloud.
