Is it possible to be an architect and interior designer?

Yes, there are many people who do both. They usually specialize in a certain type of design or focus on a specific genre of design. Still, it's totally possible to be good at both architecture and interior designing and provide clients with more comprehensive services. There is also such thing as an "architecture assistant." Architecture assistants work closely with qualified architects and are often brought into the project after initial architectural drawings have been completed to create the more detailed plans that will eventually become blueprints. These assistants can plan out element placement, furniture arrangements, lighting layouts and color schemes for various locations within the designed space.

Absolutely. The requirements for each profession are the same. Architectural skills can be put to good use when designing interiors, and vice versa.
Architects work with forces like gravity, wind loads, dynamic loads, seismic loads and structural integrity that some designers may never need to think about in their entire life. Designers also have a much more creative role where they don't just sit down at a computer drawing out blueprints from scratch while architects first have to come up with very specific plans that meet all of the necessary requirements in order to make it approved by every applicable engineer or official authority approving the design before they can go ahead with it.

Yes. The field of interior design is focused largely on the planning, furnishing and decoration of indoor (domestic) spaces or commercial ones. Architects have training in a broad range of skills - they are often skilled in the building sciences, spatial analysis, and graphic design as well as the design of buildings. Interior designers focus on smaller-scale projects with more time to devote to thorough research than architects might have for a permanent building project-- think restaurants instead of office towers.

Is it possible to be an architect and interior designer?
Yes, but you would have to become licensed as both an architect and an interior designer. This requires additional education, training, and certification. The Architectural Institute of America offers a Master of Architecture degree course for the architectural profession, or students can enroll in the Interior Design program at their choice of school to earn licensure for that profession. You do not need separate qualifications in both practices because they are mutually required by law!

Yes. Both positions can be under the same umbrella of a "designer" or "artist," and is both considered a vocational career.
Designing buildings or rooms is essentially the same as designing any other object or experience, which is sometimes called Art Design (though this term could also apply to fashion designers). Architects are typically responsible for drawing plans and sketches on paper before creating prototypical structures such as building models. A good example of their work would be to design a school - they create plans that follow architectural principles and then prospective students will give feedback about what kinds of features they want included in the infrastructure, if applicable in that context.

The qualifying word is "architect". Interior design, on the other hand, is not a profession with clearly defined requirements. I'm certain the OP enjoys design work and can pursue it in conjunction with architecture, but they should be very careful before labeling themselves an interior architect.
While they may have studied fields such as color theory or graphics software along the way, to become an interior designer you will need to gain some outside certification or demonstrate proficiency in these areas through your portfolio. This process could take from three months to two years depending on how you track down these certifications and education opportunities. Remember that there are wrinkles that arise all of the time in regards to licensure; this article covers many of them wonderfully.