Hi Mateusz! Maggie, Shopify Partner here 
Great looking store - and adorable products. The beginnings are always slow - also bear in mind that it takes up to three months for Google to start crawling new websites. You are in a very competitive niche, and you are probably on the very back pages of Google, when someone searches for the “pet clothes” or similar keywords. People rarely go beyond 1st page of Google, which is scary thought, but that’s true. So unless you are on the 1st page for your target keyword, you might get only a bunch of visits.
The good news is that you are not relying on ads - well done. As you see, they are costly, and not always send converting traffic.
When it comes to SEO, it’s a good practice to optimize your website content, as you do, as well as work on off-page SEO, such as backlinks, social media marketing, sharing your products and blog posts with your audience.
Don’t rely on Google in sending you traffic though - make sure to actively share your website in order to bring the organic and relevant traffic yourself.
You have defined niche, which is perfect. It will be easier to find the most relevant target audience. As you are in the UK, you can target pet lovers here. Join the Facebook discussion groups for the UK pet lovers and join in the conversations. Make sure to contribute to the posts, and share your website content when appropriate.
What I like to do, is to join the group and follow the discussions - if I see a topic that repeats often [such as in your case it could be: how to make your dog more visible when taking it to the park in the evening] - you could create a blog post about the reflective dog collars or dog jackets, publish it on your website, and then share the content in the discussion, inviting people to read the post on your website.
That’s a great way to drive highly converting traffic to your site.
At the same time you need to make sure the post converts buyers into shoppers - or you might end up with high traffic, but people will only be reading the blog. Make sure to link from the blog post to your products [such as reflective jacket, if you have one], and also have a button at the end of the post asking people to sign up to receive the latest news on how to keep the dog safe etc.
Then, you just have to send regular updates to stay in touch and keep people informed of the new arrivals etc.
That’s a great start to get more organic and relevant traffic.
There are other ways as well, such as social media posting, guest blogging, influencer marketing and more.
If you don’t have enough time, or you’re not sure how to get started, consider investing in the SEO Monthly packages. These are designed specifically to optimize the website content for the most profitable keywords, and actively bring traffic that converts. SEO Monthly packages start from $50 USD per month and you can order your package today here: https://www.rockpapercopy.com/seo/
Best regards,
Maggie
Shopify Partner | Ecommerce expert
Rock Paper Copy
maggie@rockpapercopy.info