In SharePoint Online it is not possible to lookup a choice field. This can be resolved with the following workarround.
Get the column named "Company Type" which is of the "Choice" type to appear on projects.
Create a column on the Company list of the "Calculated" type and set it to indicate the value of the choice field.
Then create a lookup field on the Project list to lookup the value of the calculated field in Company list or add it to a column which already is looking something up from the company list (eg. Company).
On the Project list click List settings > Company
And tick the calculated column from the Company list
A video of the configuration can also be found here: