Will choosing the ".Store" domains for my store will affect ranking?

Yes, if you look technically, then there's no downside to SEO if you use the .store TLD.

However, as @andreabosoni mentioned, it can affect your brand, and your brand can affect your SEO.

Google does track the number of clicks that you get if you're e.g. placed as the first result in the first search page. There can be a couple of ways this could backfire:

1. A user sees your domain in the search result, and decides not to click to your site. Google notices that you were not clicked, and eventually penalises your first position if this happens frequently enough.

2. A user clicks to your site, but bounces when they see your domain/brand, and goes to some other search result on the first page. This may too be an indication to Google that your site did not provide the experience that the users were looking for, and it might affect your ranking.

So there's a human factor included that has effect in the search rankings, and you should be aware of it. Google tries to give the best results to any search that the user makes, so just make sure you're the best possible result, and Google will reward you.

Perception depends heavily on your target market. If you're targetting developers or startups, ".io" is probably almost as good as ".com". In that kind of situation, the TLD becomes more of a tiebreaker. indiehackers.com is better than indiehackers.io, but indiehackers.io is probably better than findindiehackers.com.

If you're targetting mom and pop shops that aren't heavy internet users, ".io" might be just as spammy-looking to them as ".store". For them, I would choose widgetstore.com over widget.io or widget.store every time.

I don't think you'll win anything by trying to stand out on your TLD -- as @andreabosoni said, your brand will be affected negatively from a trust perspective.