You can do this. Use VariantReference
and then VariantField
or VariantToken
items and display tag name or field value:

Thanks to a number of extension points in SXA it's fairly easy to implement a custom solution for this.
There is resolveVariantTokens
pipeline which contains, for example, such processors:
<resolveVariantTokens>
<processor type="Sitecore.XA.Foundation.Variants.Abstractions.Pipelines.ResolveVariantTokens.ResolveIFileTypeIcon, Sitecore.XA.Foundation.Variants.Abstractions" resolve="true" />
<processor type="Sitecore.XA.Foundation.Variants.Abstractions.Pipelines.ResolveVariantTokens.ResolveItemId, Sitecore.XA.Foundation.Variants.Abstractions" resolve="true" />
<processor type="Sitecore.XA.Foundation.Variants.Abstractions.Pipelines.ResolveVariantTokens.ResolveItemName, Sitecore.XA.Foundation.Variants.Abstractions" resolve="true" />
<processor type="Sitecore.XA.Foundation.Variants.Abstractions.Pipelines.ResolveVariantTokens.ResolveSize, Sitecore.XA.Foundation.Variants.Abstractions" resolve="true" />
</resolveVariantTokens>
Investigate one or two of them. You will notice that it's quite easy to add your own token called for example $tags
and then use it in Token variant item like this:

Your custom processor would simply gets tags out of an item and just return tags name or <span class="tag-something>TAG_NAME</span>
.