I don’t have much in savings but does my house (or my car) count as an asset that will affect whether I am eligible to claim?

Assets are not taken into account when claiming Universal Credit, but savings of £6,000 and up are. Owning your own home and car will not stop you claiming UC. In fact, there may be some extra benefits you can apply for as a part of the Universal Credit claim.

This info is from the government in relation to UC:

Owner-occupiers who are leaseholders can get help with service charges paid as a housing element within Universal Credit. Although you normally have to serve a waiting period of approximately nine months from the date of your Universal Credit claim before you can receive this help.

Help with mortgage interest payments is available in the form of a loan on which you will be charged interest. This is separate from your claim for Universal Credit. Please read our Support for Mortgage Interest Loan guide for more detailed information about this loan

Read more here: https://www.turn2us.org.uk/Benefit-guides/Universal-Credit-housing-costs/How-much-will-I-get-I-m-a-homeowner#guide-content